<^ 


\I^ 


V 


ELEMENTS 


.Analytic  Geometry. 


BY 

G.  A.  WENTWORTH,  A.M., 

PROFESSOR   OF    MATUE.MATKS    IN    VUILLU'S    EXETER   ACADEMY. 


I 


BOSTON: 

PUBLISHED   BY   GINN   &   COMPANY. 

188G. 


Entered,  according  to  the  Act  of  Congress,  in  the  year  1886,  by 

G.  A.  WEXTWORTH, 

in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


J.  S.  CusuiNG  &  Co.,  Pkinteks,  Boston. 


PREFACE. 


rpHIS  book  is  intended  for  beginners.  As  beginners  generally  find 
great  difficulty  in  comprehending  the  connection  between  a  locus 
and  its  equation,  the  opening  chapter  is  devoted  mainly  to  an  attempt, 
by  means  of  easy  illustrations  and  examples,  to  make  this  connection 
clear. 

Each  chapter  abounds  in  exercises ;  for  it  is  only  by  solving 
problems  which  require  some  degree  of  original  thought  that  any 
real  mastery  of  the  study  can  be  gained. 

The  more  difficult  propositions  have  been  put  at  the  ends  of  the 
chapters,  under  the  heading  of  "Supplementary  Propositions."  This 
arrangement  makes  it  possible  for  every  teacher  to  mark  out  his  own 
course.  The  simplest  course  will  be  Chapters  I.-III.  and  Chapters 
V.-VII.,  with  Review  Exercises  and  Supplementary  Propositions  left 
out.  Between  this  course  and  the  entire  work  the  teacher  can  exer- 
cise his  choice,  and  take  just  so  much  as  time  and  circumstances  will 
allow. 

The  author  has  gathered  his  materials  from  many  sources,  but  he 
is  particularly  indebted  to  the  English  treatise  of  Charles  Smith. 
Special  acknowledgment  is  due  to  G.  A.  Hill,  A.M.,  of  Cambridge, 
Mass.,  for  assistance  in   the  preparation   of  the   book. 

Corrections  and   suggestions  will  be  thankfully  received. 

G.  A.  WENTWORTH. 
Phillips  Exeter  Academy, 
July,  1886. 


184022 


CONTENTS. 


CHAPTER   I.     Loci  and  their  Equations. 


PAOB. 


1-4.   Preliminary  Notions 1 

5.    Circular  Measure  of  an  Angle 5 

6-7.    Distance  between  Two  Points 6 

8-9.    Division  of  a  Line 8 

10-16.   Constants  and  Variables 10 

17-24.    Locus  of  an  Equation 14 

25-29.    Intersections  of  Loci 22 

30-34.   Construction  of  Loci 25 

35.   Equation  of  a  Curve 31 

Review  Exercises .•    .  33 

CHAPTER  n.    The  Straight  Line. 

36-41.    Equations  of  the  Straight  Line 36 

42-47.   General  Equation  of  the  First  Degree 43 

48-51.    Parallels  and  Perpendiculars 47 

52-53.   Angles 50 

54-56.    Distances 54 

57-58.   Areas       59 

Review  Exercises 62 

59-67.   Supplementary  Propositions 67 

CHAPTER   in.     The  Circle. 

68-70.    Equations  of  the  Circle 77 

71-75.    Tangents  and  Normals 83 

Review  Exercises 90 

76-87.   Supplementary  Propositions 95 


VI  CONTENTS. 


CHAPTER   IV.     Different  Systems  of  Co-ordinates. 

SKCTIONS.  p^eH 

88-90.    Oblique  Co-ordinates 105 

91-92.    Polar  Co-ordinates 107 

93-100.    Transformation  of  Co-ordinates HI 

Review  Exercises 117 


CHAPTER  V.     The  Parabola. 

101-108.   The  Equation  of  the  Parabola 119 

109-112.    Tangents  and  Normals 126 

Review  Exercises 131 

113-127.   Supplementary  Propositions 134 


CHAPTER  VI.     The  Ellipse. 

128-140.    Simple  Properties  of  the  Ellipse 145 

141-147.    Tangents  and  Normals 155 

Review  Exercises  .     .  _ 161 

148-164.    Supplementary  Propositions 163 


CHAPTER  VII.     The  Hyperbola. 

165-173.   Simple  Properties  of  the  Hyperbola 180 

17-4-179.    Tangents  and  Normals 187 

Review  Exercises 189 

180-197.   Supplementary  Propositions 190 


CHAPTER   VIII.     Loci  of  the  Second  Order. 

198-200.   General  Equation  of  the  Second  Degree 201 

201-202.    Loci  of  the  Second  Order  that  have  a  Finite  Centre    .     .  207 

203.  Loci  of  the  Second  Order  that  have  no  Finite  Centre  .     .  210 

204.  Special  Cases  of  Loci 215 

205.  Summary 217 

206-208.    Examples  and  Definitions 217 

Exercises •.     .     .     .  221 


AJN^ALYTIC   GEOMETRY, 


CHAPTER   I. 


LOCI    AND    THEIR    EQUATIONS. 

Preliminary  Notions. 

1.  Let  XX'  and  YY'  (Fig.  1)  be  two  fixed  lines  perpen- 
dicular to  each  other,  and  intersecting  in  a  point  0.  These 
lines  divide  the  plane  in  which  they  lie  into  four  similar  parts. 

■n 


E 

P 

P,  F 

X' 

IN 

0 

M 

K 

IB 

T 

3... 

v 

Fig.   I. 


A  H 


Let  these  parts  be  called  Quadrants  (as  in  Trigonometry),  and 
^distinguished  by  naming  the  area  between  OX  and  OF  the 
\  first  quadrant ;  that  between  0  Y  and  OX'  the  second  quad- 
rant ;  that  between  OX'  and  0  F'  the  third  quadrant ;  and 
that  between  OY'  and  OX i\ie  fourth  quadrant. 

Suppose  the  position  of  a  point  is  described  by  saying  that  its 


2  ANALYTIC   GEOMETRY. 

distance  from  YY',  expressed  in  terms  of  some  chosen  unit 
of  length,  is  3,  and  its  distance  from  JTX'  is  4.  It  is  clear 
that  in  each  quadrant  there  is  one  point,  and  only  one,  which 
will  satisfy  these  conditions.  The  position  of  the  point  in 
each  quadrant  may  be  found  by  drawing  parallels  to  YY' 
at  the  distance  3  from  YY\  and  parallels  to  XX'  at  the  dis- 
tance 4  from  XX' ;  then  the  intersections  Pj,  F^,  Pz,  and  P^ 
satisfy  the  given  conditions. 


:l!>            , 

T 

,B 

E 

■f. 

^F 

X' 

\N 

0 

M:        X 

..i^....„. 

T 

\l\ 

G 

\c 

Y' 

&  H 

Fig.  I. 


2.  In  order  to  determine  which  one  of  the  four  points, 
Pi,  P-i,  P3,  Pi,  is  meant,  we  adopt  the  rule  that  opposite 
directions  shall  be  indicated  by  unlike  signs.  As  in  Trigo- 
nometry, distances  measured  from  YY'  to  the  right  are  con- 
sidered positive ;  to  the  left,  negative.  Distances  measured  from 
XX'  upward  are  positive;  downward,  negative.  Then  the 
position  of  Pi  will  be  denoted  by  +3,  +4  ;  of  P„  by  -3,  +4 ; 
of  P3,  by -3, -4;  of  P„  by +3, -4. 


3.  This  method  of  determining  the  position  of  a  point  in  a 
plane  is  the  method  commonly  employed  in  Analytic  Geom- 
etry. It  enables  us  to  represent  position  by  numbers  ;  and^ 
by  reasoning  with  these  numbers,  to  investigate  the  properties' 
of  geometrical  figures.  The  science  of  Analytic  Geometry 
consists  of  investigations  of  this  kind. 


LOCI    AND    THEIR    EQUATIONS.  6 

Note.  The  first  man  who  employed  this  method  successfully  in 
investigating  the  j>roperties  of  certain  figures  was  tlie  Fre«ch  jihiloso- 
pher  Descartes,  whose  work  on  Geometry  appeared  in  the  year  1(137. 


Co-ordinates  ;  XX'  is  called  the  Axis  of  Abscissas,  or  Axis  of  x ; 
YY',  the  Axis  of  Ordinates,  or  Axis  of  y.  The  intersection  0  is 
called  the  Origin. 

The  two  distances  (with  signs  prefixed)  which  determine 
the  position  of  a  point  are  called  the  Co-ordinates  of  the  point ; 
the  distance  of  the  point  from  YY'  is  called  its  Abscissa;  and 
the  distance  from  XX',  its  Ordinate. 

The  letters  x  and  ?/  are  in  common  use  as  general  symbols 
or  abbreviations  for  the  words  "abscissa"  and  "ordinate" 
respectively.  For  the  sake  of  brevity,  a  point  is  often  rep- 
resented algebraically  by  simply  writing  the  values  of  its 
co-ordinates  within  brackets,  the  value  of  the  abscisea  being 
always  written  first. 

Thus  Pi  (Fig.  1)  is  the  point  (3,  4),  F,  the  point  (-3,  4), 
Ps  the  point  (— 3,'^^4),  and  P4  the  point  (3,  —4).  In  gen- 
eral the  point  whose  co-ordinates  are  x  and  7/  is  the  point 

(^,y)- 

/  Ex.  1. 

1.  "What* are  the  co-Qi*dinates  of  the  origin  ? 

2.  In  what  cjuadrants  are  the  following  points  (a  and  b 
being  given  lengths) : 

(~-a,-h),:      (-a,h),       (a,b),       {a,-b). 

3.  To  what  quadrants  is  a  point  limited  if  its  abscissa  is 
positive  ?   negative  ?   ordinate  positive  ?    ordinate  negative  ? 

*4.  In  what  line  does  a  point  lie  if  its. abscissa  =  0?  if  its 
ordinate  =  0  ? 

5.  A  point  {x,y)  moves  parallel  to  the  axis  of  .r ;  which  one 
of  its  co-ordinates  remains  constant  in  value  ? 


4  ANALYTIC    GEOMETRY, 

6.  Construct  or p/onhe  points:  (2,3),  (3,-3),  (-1,-3), 
(-4,4),  (3,0),  (-3,0),  (0,4),  (0,-1),  (0,0). 

Note.  To  plot  a  point  is  to  mark  its  proper  position  on  paper,  when 
its  co-ordinates  are  given.  The  first  thing  to  do  is  to  draw  the  two 
axes.     The  rest  of  the  work  is  obvious  after  a  study  of  Nos.  1-3. 

7.  Construct  the  triangle  whose  vertices  are  the  points 
(2,4),  (-2,7),  (-6,-8). 

8.  Construct  the  quadrilateral  whose  vertices  are  the  points 
(7,2),  (0,-9),  (-3,-1),  (-6,4). 

9.  Construct  the  quadrilateral  whose  vertices  are  (—3,  6), 
(-3,  0),  (3,  0),  (3,  6).     What  kind  of  a  quadrilateral  is  it  ? 

10.  Mark  the  four  points  (2,  1),  (4,  3),  (2,  5),  and  (0,  3), 
and  connect  them  by  straight  lines.  What  kind  of  a  figure 
do  these  four  lines  enclose? 

11.  The  side  of  a  square  = 'a;  the  origin  of  co-ordinates 
is  the  intersection  of  the  diagonals.  What  are  the  co-ordinates 
of  the  vertices  (i.)  if  the  axes  are  parallel  to  the  sides  of  the 
square  ?      (li.)  if 'the  axes  coincide  with  the  diagonals  ? 

^-(..)(f  f)  (-1  2}  (-J -5}  (I -^)^ 

(i,.)(2V2,   0),    (0,   ?vl),    (^-|v/2,   0),    (0,   -2V2]. 

12.  The  side  of  an  equilateral  triangle  =  a  ;  the  origin  is 
taken  at  one  vertex  and  the  axis  of  x  coincides  with  one  side. 
What  are  the  co-ordinates  of  the  three  vertices  ? 

Ans.   (0,0),    (a,  0),    K    f  VSY 

13.  The  line  joining  two  points  is  bisected  at  the 
If  the  co-ordinates  of  one  of  the  points  are  a  and  b,  what 
the  co-ordinates  of  the  other  ? 

14.  Connect  the  points  (5,  3)  and  (5,  —  3)  by  a  straight 
line.     What  is  the  direction  of  this  line  ?  *' 

./|V 


origin, 
are 


\ 


LOCI    AND    THEIR    EQUATIONS. 


Circular  Measure. 


5.  In  Analytic  Geometry,  angles  are  often  expressed  in 
degrees,  minutes,  and  seconds;  but  sometimes  it  is  very' con- 
venient to  employ  the  Circular  Pleasure  of  an  angle. 

In*  circular  measure  an  angle  is  defined  by  the  equation 

1  arc 

angle  =  — - — . 
radius 

in  which  the  word  "  arc  "  denotes  the  length  of  the  arc  corre- 
sponding to  the  angle  when  both  arc  and  radius  are  expressed 
in  terms  of  a  common  linear  unit. 

This  equation  gives  us  a  correct  measure  of  angular  magni- 
tude, because  (as  shown  in  Geometry)  for  a  given  angle  the 
value  of  the  above  ratio  of  arc  and  radius  is  constant  for  all 
values  of  the  radius.' 

If  the  radius  =  1,  the  equation  becomes 

angle  =  arc  ;  that  is, 

In  circular  measure  an  angle  is  measured  hy  the  length  of 
the  arc  suhtended  hy  it  in  a  unit  circle. 

It  is  shown  in  Geometry  that  the  circumference  of  a  unit 
circle  =  . 2 tt;  as  this  circumference  contains  3G0°  common 
measure,  the  two  measures  are  easily  compared  by  means 
of  the  relation 

360  degrees  =  2  TT  units,  circular  measure. 

Ex.  2. 

1.  Find  the  value  in  circular  measure  of  the  angles  1®, 
45°,  90°,  180°,  270°. 

180'   ^'''    ^'''  ''' 

2.  In  circular  measure,  unit  angle  is  that  angle  whose  arc 

is  equal  to  the  radius  of  the  circle.     What  is  the  value  of  this 

angle  in  degrees,  etc.  ? 

Am.   57°  W  45^^ 


ANALYTIC   GEOMETRY. 


Distance  between  Two  Points. 

6.   To  find  the  distance  between  two  given  points. 

Let  P  and  Q  (Fig.  2)  be  the  given  points,  x^  and  3/1  the 
co-ordinates  of  P,  x,  and  3/2  those  of  Q.  Also  let  d=  PQ  = 
the  required  distance. 


3f 


Fig.  2. 


iS^X 


Traw  PJ/and  Qil^-L  to  OX,  and  PP  II  to  OX. 

Then  0J/=  .Tj,  MP  =^  y,, 

ON=x„  m  =  y2f 

PR  =  X2  —  Xu       QR  =  y.,  -  2/1. 

By  Geometry, 

c^2  ==  (^x.,  -  x,y  +  (y,  -  y,y  ; 

whence  d  =  ^{x.^  —  a?i)"-^  -ff  2/2  —  Z/i)'-^-  [1] 

Since  (a^i  —  a'2)^  =  (^2  —  ^l)^  it  makes  no  difference  which 
point  is  called  (.Ti,  3/1),  and  which  (0:2, 3/2). 

7.  Equation  [1]  is  perfectly  general,  holding  true  for  points 
situated  in  any  quadrant.  Thus,  if  P  be  in  the  second  quad- 
rant and  Q  in  the  third  quadrant  (Fig.  3),  x^—x^  is  obvi- 
ously equal  to  the  leg  RQ  ;  and  since  3/2  is  negative,  3/2  —  2/1  is 
the  sum  of  two  negative  numbers,  and  is  equal  to  the  absolute 
length  of  the  leg  RP  with  the  —  sign  prefixed. 


LOCI    AND    TUEIR    EQUATIONS.  7 

Note.  The  learner  should  satisfy  himself  that  equation  [1]  is  perfectly- 
general,  by  constructing  other  special  cases  in  which  the  points  Pand  Q 
are  in  different  quadrants.  In  every  case  he  will  find  that  the  numeri- 
cal values  of  the  expressions  (-Tj  —  a-^)  and  (ya~2/i)  ^-re  the  legs  of  the 
right  triangle  the  hypotenuse  of  which  is  the  required  distance  PQ. 

Equation  [1]  is  merely  one  illustration  of  a  general  truth,  of  which 
the  learner  will  gradually  become  convinced  as  he  proceeds  with  the 
stud}'-  of  the  subject ;  namely,  that  theorems  and  formulas  deduced  by 
reasoning  with  points  or  lines  in  the  first  quadrant  {where  the  co-ordinates 
are  always  positive)  must,  from  the  very  nature  of  the  analytic  method, 
hold  true  when  the  points  or  lines  are  situated  in  the  other  quadrants. 

Ex.  3. 

Find  the  distance  z 

1.  From  the  point  (—2,  5)  to  tile  point  (—8,  —3). 

2.  From  the  point  (1,  3)  to  the  point  (6,  15). 

3.  From  the  point  (—4,  5)  to  the  point  (0,  —2). 

4.  From  the  origin  to  the  point  (—6,  —8). 

5.  From  the  point  (a,  b)  to  the  point  (—a,  —h). 

Find  the  lengths  of  the  sides  of  a  triangle 

6.  If  the  vertices  are  the  points  (15,-4),  (-9,3),  (11,24). 

7.  If  the  vertices  are  the  points  (2,  3),  (4,  -5),  (-3,  -6). 

8.  If  the  vertices  are  the  points  (0,  0),  (3,  4),  (-3,  4). 

9.  If  the  vertices  are  the  points  (0,  0),  (--a,  0),  (0,  —h). 

10.  The  vertices  of  a  quadrilateral  are  (5,  2),  (3,  7),  (—1,  4), 
(—3,  —2).  Find  the  lengths  of  the  sides  and  also  of  the 
diagonals. 

11.  One  end  of  a  line  whose  length  is  13  is  the  point 
(—4,  8)  ;  the  ordinate  of  the  other  end  is  3.  What  is  its 
abscissa? 

12.  What  equation  must  the  co-ordinates  of  the  point  {x,  y) 
satisfy  ii'its  distance  from  the  point  (7,  —2)  is  equal  to  11  ? 


8 


ANALYTIC    GEOMETRY. 


13.  What  equation  expresses  algebraically  the  fact  that 
the  point  (x,  y)  is  equidistant  from  the  points  (2,  3)  and 
(4,5)? 

14.  If  the  value  of  a  quantity  depends  on  the  sqiiare  of  a 
length,  it  is  immaterial  whether  the  length  be  considered 
positive  or  negative.     Why  ? 


Division  of  a  Line. 
8.    To  bisect  the  line  joining  two  given  points. 

Let  P  and  Q  (Fig.  4)  be  the  given  points  {x^,  y^)  and  {x^.y-^. 
Let  X  and  y  be  the  co-ordinates  of  i?,  the  mid-point  of  PQ. 

The  meaning  of  the  problem  is  to  find  the  values  of  x  and  y 
in  terms  of  x^^,  yi,  and  x^,  y%. 


Y 

pZ 

f( 

I 

: 

1 

0 

M 

a 

.Y 

X 

Y 

/« 

/ 

i 

p/ 

/ 

A 

\ 

1 

0 

M 

< 

5    NX 

Fig.  4. 


Fig.  5. 


Draw  P3f,  PS,  QN 1.  to  0X\  also  draw  PA,  PB  \\  to  OX. 
Then  rt.  A  PPA  =  rt.  A  PQB  (hypotenuse  and  one  acute 
angle  equal). 

Therefore  PA  =  PB,   and   AP  =  BQ\ 

also  218  =SN. 

By  substitution,    x  —  Ti  -—  X2 

whence  x  =  — ^-^^ — '-  5    y  = 


X,    and   y  —  y^  =  y^  —  y, 


Vx  -^1/2 


[2] 


LOCI    AND    THEIR    F.QUATIONS.  9 

9.  To  divide  the  line  joining  two  given  points  into  two  imi  is 
having  a  given  ratio  ni :  n. 

Let  Pand  Q  (Fig.  5)  be  the  given  points  {i\,  ?/,)  and  (x^,^^)- 
Let  H  be  the  required  point,  such  that  Pli  :  RQ  —  ni:  n,  and 
let  X  and  y  denote  the  co-ordinates  of  B. 

Complete  the  figure  by  drawing  lines  as  in  Fig.  4. 

The  rt.  A  PEA  and  EQB,  being  mutually  equiangular, 
are  similar  ;  therefore 

BB      BQ      n      '^"^      5q      BQ^  n 
Substituting  for  the  lines  their  values,  we  have 

^~^^._  ^-^^      and      y~y^__  ^^^ 
x-i  —  X      n  Vi  —  y      ^ 

Solving  these  equations  for  a:  and  y,  we  obtain 

_  mx,  +  nx^         _  my,  +  ny^ 

If  m  =  ?2,  we  have  the  special  case  of  bisecting  a  line  already 
considered  ;  and  it  is  easy  to  see  that  the  values  of  x  and  y 
reduce  to  the  forms  given  in  [2]. 

X.  4. 

"What  are  the  co-ordinates  of  the  point 

1.  Half-way  between,  the  points  (5,  3)  and  (7,  9)  ? 

2.  Half-way  between  the  points  (-6,  2)  and  (4,  -2)  ? 

3.  Half-way  between  the  points  (5,  0)  and  (-1,  -4)  ? 

4.  The  vertices  of  a  triangle  are  (2,  3),  (4,  ~5),  (-3,  -6)  ; 
find  the  middle  points  of  its  sides. 

5.  The  middle  point  of  a  line  is  (6,  4),  and  one  end  of  the 
lineis  (5,  7).     What  are  the  co-ordinates  of  the  other  end  ? 

6.  A  line  is  bisected  at  the  origin  ;  one  end  of  the  line  is  the 
point  {—a,  b).    What  are  the  co-ordinates  of  the  other  end  ? 


10  ANALYTIC    GEOMETRY. 

7.  Prove  that  the  middle  point  of  the  hypotenuse  of  a  right 
triangle  is  equidistant  from  the  three  vertices. 

8.  Prove  that  the  diagonals  of  a  parallelogram  mutually 
bisect  each  other. 

9.  Show  that  the  values  of  x  and  y  in  [2]  hold  true  when 
the  two  given  points  both  lie  in  the  second  quadrant. 

10.  Solve  the  problem  of  §  9  when  the  line  FQ  is  cut 
externally  instead  of  internally,  in  the  ratio  m  :  n. 

11.  What  are  the  co-ordinates  of  the  point  which  divides 
the  line  joining  (3,-1)  and  (10,  6)  in  the  ratio  3:4? 

12.  The  line  joining  (2,  3)  and  (4,  —5)  is  trisected  ;  deter- 
mine the  point  of  trisection  nearest  (2,  3). 

13.  A  line  AB  is  produced  to  a  point  C,  such  that  i?(7  = 
^AB.  If  A  and  B  are  the  points  (5,  6)  and  (7,  2),  what  are 
the  co-ordinates  of  C? 

14.  A  line  AB  is  produced  to  a  point  C,  such  that  AB  :  BC 
=  4:7.  If  ^  and  B  are  the  points  (5,  4)  and  (6,  -9),  what 
are  the  co-ordinates  of  (7? 

15.  Three  vertices  of  a  parallelogram  are  (1,  2),  (—5,  —3), 
(7,  -6).     What  is  the  fourth  vertex  ? 

Constants  and  Variables. 

10.  In  Analytic  Geometry  a  line  is  regarded  as  a  geometric 
magnitude  traced  or  generated  by  a  moving  point, — just  as  we 
trace  on  paper  what  serves  to  represent  a  line  to  the  eye  by 
moving  the  point  of  a  pen  or  pencil  over  the  paper. 

We  shall  find  that  great  advantages  are  to  be  gained  by 
defining  a  line  in  this  way,  but  we  must  be  prepared  from 
the  outset  to  make  an  important  distinction  in  the  use  of 
symbols  representing  lengths.  We  must  distinguish  between 
symbols  which  denote  definite  or  fixed  lengths  and  those 
which  denote  variable  lengths. 


LOCI    AND    THEIR    EQUATIONS. 


11 


11.  A  simple  example  will  serve  to  illustrate  this  difference. 
Let  A  (Fig.  6)  be  the  point  (3,  4).  Then  OA  =  V9  +  16  =  5. 
Now  let  a  point  F  describe  the  line  OA  by  moving  from  0 
to  A,  and  let  the  co-ordinates  of  P  be  denoted  by  x  and  y ; 
also  let  z  denote  the  length  OF  at  any  position  of  F.  Then 
it  is  clear  that  the  distance  OF  or  z  will  be  equal  to  0,  to 
begin  with,  and  will  increase  in  value  continuously  until  it 
becomes  equal  to  5. 

F 


M        B      X 

Fig.   6. 


Here  the  word  continuoiishj  deserves  special  attention.  It 
means  that  F  must  pass  successively  through  every  conceiv- 
able position  on  the  line  OA  from  0  to  A;  that,  therefore,  z 
must  have  in  succession  every  conceivable  value  between  0 
and  5.  There  will  be  one  position  of  F  for  which  z  is 
equal  to  2 ;  there  will  be  another  position  of  F  for  which 
z  is  equal  to  2.000001 ;  but  before  reaching  this  value  it  must 
first  pass  through  all  values  between  2  and  2.000001. 

In  the  same  way  the  co-ordinates  of  F,  namely,  x  and  y, 
both  pass  through  a  continuous  series  of  changes  in  value 
unlimited-  in  number,  the  abscissa  x  increasing  continuously 
from  0  to  3,  and  the  ordinate  y  from  0  to  4. 

We  may  now  divide  the  lengths  considered  in  this  example 
into  two  classes : 

(1)  Lengths  supposed  to  remain  constant  in  value,  namely, 
the  co-ordinates  of  A  and  the  distance  OA  ;  (2)  lengths  sup- 
posed to  vary  continuously  in  value,  namely,  the  co-ordinates 
of  F,  (x  and  y),  and  the  distance  OF  or  z. 


12  ANALYTIC    GEOMETRY. 

Quantities  of  the  first  kind  in  any  problem  are  called  co7i- 
stant  quantities,  or,  more  briefly,  Constants. 

Quantities  of  the  second  kind  are  called  variable  quantities, 
or,  more  briefly.  Variables. 

12.  Two  variables  are  often  so  related  that  if  one  of  them 
changes  in  value,  the  other  also  changes  in  value.  The  second 
variable  is  then  said  to  be  a  function  of  the  first  variable. 
The  second  variable  is  also  called  the  dependent  variable, 
while  the  first  is  called  the  independent  variable.  Usually 
the  relation  between  two  variables  is  such  that  either  may  be 
treated  as  the  independent  variable,  and  the  other  as  the 
dependent  variable. 

Thus,  in  §  11,  if  ^ve  suppose  z  to  change,  then  both  x  and 
y  will  change  ;  the  values  of  x  and  y  then  will  depend  upon 
the  value  given  to  z  ;  that  is,  x  and  y  will  be  functions  of  z. 
But  we  may  also  suppose  the  value  of  x,  the  abscissa  of  P, 
to  change ;  then  it  is  clear  that  the  values  of  both  y  and  z 
must  also  change.  In  this  case  we  take  x  as  the  independent 
variable,  and  values  of  y  and  z  will  depend  upon  the  value 
of  X  ;   that  is,  y  and  z  will  hQ  functions  of  x. 

13.  The  most  concise  way  to  express  the  relations  of  the 
constants  and  variables  which  enter  into  a  problem  is  by 
means  of  algebraic  equations. 

The  co-ordinates  of  P  (Fig.  6)  throughout  its  motion  are 
always  x  and  y ;  and  the  triangle  0PM  is  similar  to  the 
triangle  OAB.     Hence,  for  any  position  of  P, 

^  =  -,  and  ^  —  x^  -^y"^  \ 
X      3  * 

whence,  by  solving, 

4  T         5 

?/  =  -X,  and  z  =  -x, 

equations  which  express  the  values  of  y  and  z,  respectively, 
in  terms  of  x  as  the  independent  variable. 


LOCI    AND    THEIR    EQUATIONS.  13 

14.  In  §  11,  instead  of  assuming  3  and  4  as  the  co-ordi- 
nates of  P,  we  might  have  employed  two  letters,  as  a  and  b, 
with  the  understanding  that  these  letters  should  denote  two 
co-ordinates  which  remain  constant  in  value  during  the  motion 
of  P.  If  we  choose  these  letters,  and  then  proceed  exactly 
as  in  §  13,  we  obtain  for  the  values  of  y  and  z, 

•^      a   '  a 

15.  There  is  a  noteworthy  difference  between  the  constants 
3  and  4  and  the  constants  a  and  b.  The  numbers  3  and  4 
are  unalterable  in  value  ;  they  cannot  be  supposed  to  change 
under  any  circumstances.  The  letters  a  and  b  are  constants 
in  this  sense  only,  that  they  do  not  change  in  value  when  we 
suppose  X  or  y  or  z  to  change  in  value ;  in  other  words,  they 
are  not  functions  of  a;  or  y  or  z  in  the  particular  problem, 
under  discussion.  In  all  other  respects  they  are  free  to  rep- 
resent as  many. different  values  as  we  choose  to  assign  to  them. 

Constants  of  the  first  kind  (arithmetical  numbers)  are  called 
absolute  constants.  Constants  of  the  second  kind  (letters)  are 
called  arbitrary  or  general  constants. 

16.  By  general  agreement,  variables  are  represented  by  the 
last  letters  of  the  alphabet,  as  x,  y,  z;  while  constants  are 
most  commonly  represented  by  the  first  letters,  a,  b,  c,  etc.,  or 
by  the  last  letters  with  subscripts  added,  as  x^,  3/1,  x^,  y^,  etc. 

Ex.  5. 

1.  A  point  P  (x,  y)  revolves  about  the  point  Q  (.Tj.  y^),  keep- 
ing always  at  the  distance  a  from  it.  Mention  the  constants 
and  the  variables  in  this  case.  What  is  the  total  change  in 
the  value  of  each  variable  ? 

2.  A  point  Q  (x,  y)  moves :  first  parallel  to  the  axis  of  y, 
then  parallel  to  the  axis  of  x,  then  equally  inclined  to  the 
axes.     Point  out  in  each  case  the  constants  and  the  variables. 


14 


ANALYTIC    GEOMETRY. 


Locus  OF  AN  Equation. 

17.  Let  us  continue  to  regard  x  and  y  as  the  co-ordinates  of 
a  point,  and  proceed  to  illustrate  the  meaning  of  an  algebraic 
equation  containing  one  or  both  of  these  letters. 

Take  as  the  first  case  the  equation  a;  — 4  =  0,  whence  a:  =  4. 
It  is  clear  that  this  equation  is  satisfied  by  the  co-ordinates  of 
every  point  so  situated  that  its  abscissa  is  equal  to  4 ;  there- 
fore it  is  satisfied  by  the  co-ordinate  of  every  point  in  the  line 


Fig.  7. 

AB  (Fig.  7),  drawn  I!  to  OY,  on  the  right  of  OF,  and  at  the 
distance  4  from  0  Y.  And  it  is  also  clear  that  this  line  con- 
tains all  the  points  whose  co-ordinates  will  satisfy  the  given 
equation. 

The  line  AB,  then,  may  be  regarded  as  the  geometric  rep- 
resentation or  meaning  of  the  equation  x  —  4  =  0;  and  con- 
versely, the  equation  x  —  4  =  0  may  be  considered  to  be  the 
algebraic  representative  of  this  particular  line. 

In  Analytic  Geometry  the  line  AB  is  called  the  locus  of 
the  equation  a;  —  4  =  0  ;  conversely,  the  equation  :r  —  4  =  0  is 
known  as  the  equation  of  the  line  AB. 

The  line  AB  is  to  be  regarded  as  extending  indefinitely  in 
both  directions.  If  AB  be  described  by  a  point  P,  moving 
parallel  to  the  axis  of  y,  then  at  all  points  x  is  constant  in 


LOCI    AND    THEIR    EQUATIONS. 


15 


value  and  equal  to  4,  while  y  (which  does  not  appear  in  the 
given  equation)  is  a  variable,  passing  through  an  unlimited 
number  of  values,  both  positive  and  negative. 

18.  The  equation  x  —  y  —  ^,ox:  x  =  y,  states  in  algebraic 
language  that  the  abscissa  of  the  point  is  always  equal  to  the 
ordinate. 


' 

Y 

77/ 

Values  of  X. 

Values  of  y. 

p/ 

^ 

0     .     .     . 

.      .        0. 

1   .   .   . 

.      .        1. 

/ 

2    .     .     . 

.     .       2. 

/ 

0 

X 

1—1 
1 

.     .  -1. 

/ 

etc. 

etc. 

/ 

Fig.  8. 


If  we  draw  through  the  origin  0  (Fig.  8)  a  straight  line 
AB,  bisecting  the  first  and  third  quadrants,  then  it  is  easy  to 
see  that  the  given  equation  is  satisfied  hy  every  'point  in  this 
line  and  hy  no  other  points.  If  we  conceive  a  point  P  to 
move  so  that  its  abscissa  shall  always  be  equal  to  its  ordinate, 
then  the  point  must  describe  the  line  AB.  In  other  w^ords, 
if  the  point  P  is  obliged  to  move  so  that  its  co-ordinates 
(which  of  course  are  variables)  shall  always  satisfy  the  con- 
dition expressed  by  the  equation  a;  —  y  =  0  ;  then  the  motion 
of  P  is  confined  to  the  line  AB. 

The  line  AB  is  the  locus  of  the  equation  x  —  y  =  0,  and 
this  equation  represents  the  line  AB. 

19.  The  equation  20:  +  ?/— 3  =  0  is  satisfied  by  an  unlimited 
number  of  values  of  x  and  y.  We  may  find  as  many  of  them 
as  we  please  by  assuming  values  for  one  of  the  variables,  and 
computing  the  corresponding  values  of  the  other. 


16 


ANALYTIC    GEOMETRY. 


If  we  assume  for  x  the  values  given  below,  we  easily  find 
for  y  the  corresponding  values  given  in  the  next  column. 


Values  of  x.  Values  of  y. 

0 3. 

1 1. 

2 -1. 

o  o 

O —  'J. 

4 -5. 

-1 5. 

-2 7. 

-3 9. 

-4 11. 

etc.  etc. 


Fig.  9. 


Plotting  these  points  (as  shown  in  Fig.  9),  we  obtain  a  series 
of  points  so  placed  that  their  co-ordinates  all  satisfy  the  given 
equation.  By  assuming  for  x  values  between  0  and  1,  1  and 
2,  etc.,  ^ve  might  in  the  same  way  obtain  as  many  points  as 
we  please  between  A  and  JB,  B  and  C,  etc.  In  this  case, 
however,  the  points  all  lie  in  a  straight  line  (as  will  be  shown 
later)  ;  so  that  if  any  tiuo  points  are  found,  the  straight  line 
drawn  through  them  will  include  all  the  points  whose  co-ordi- 
nates satisfy  the  given  equation.  Now  imagine  that  a  point  P, 
the  co-ordinates  of  w^hich  are  denoted  by  x  and  y,  is  required 
to  move  in  such  a  way  that  the  values  of  x  and  y  shall  always 
satisfy  the  equation  2x-\-y  —  ^  =  0-,  then  P  must  describe 
the  line  AB,  and  cannot  describe  any  other  line. 

The  line  AB  is  the  locus  of  the  equation  2x-\-  y  —  o  =  0. 

20.  Thus  far  we  have  taken  equations  of  the  first  degree. 
Let  us  now  consider  the  equation  x^  —  y'^  =  0.  By  solving  for 
?/,  we  obtain  y  =  d=  a;.     Hence  for  every  value  of  x  there  are 


LOCI    AND    THEIR    EQUATIONS. 


17 


two  values  of  y,  both  equal  numerically  to  x,  but  having  unlike 
signs.  Thus,  for  assumed  values  of  x,  we  have  corresponding 
values  of  y  given  below  : 

Vahies  of  x.  Values  of  y. 

0 0. 

1 
o 

3 

1 

o  

Fig.   10. 


1, 

-1. 

o 

—  2. 

3, 

-3. 

1, 

1. 

2 

9 

3, 

3. 

zK 

V 

/ 

/b 

/ 

\ 

X 

A^ 

\ 

By  plotting  a  few  points,  and  comparing  this  case  with  the 
example  in  §  19,  it  becomes  evident  that  the  locus  of  the 
equation  consists  of  two  lines,  AB,  CD  (Fig.  10),  drawn 
through  the  origin  so  as  to  bisect  the  four  quadrants. 

21.  There  is  another  way  of  looking  at  this  case.  The  equa- 
tion x-—y'^  =  0,  by  factoring,  may  be  written  {x—y)  (:r+y)  =  0. 
Now  the  equation  is  satisfied  if  either  factor  =  0 ;  hence,  it  is 
satisfied  if  x  —  y  =  Q,  and  also  if  x-\-y  =  0.  We  know  (see 
§  19)  that  the  locus  of  the  equation  x  —  y  =  0  is  the.  line 
AB  (Fig.  8).  And  the  locus  of  the  equation  x-\-y  =  0  (or 
x  —  ~y^  is  evidently  the  line  CD,  since  every  point  in  it  is 
so  placed  that  the  two  co-ordinates  are  equal  numerically  but 
unlike  in  sign.  Therefore  the  original  equation  x^  —  y'^  =  0 
is  represented  by  the  pair  of  lines  AB  and  CD  (Fig.  10). 


22.    Let  us  next  consider  the  equation  x^ 
ing  for  y,  we  obtain  y  =  ±V25  — :? 


7/^=25.     Solv- 
When  rr  <  5  there 


are  two  values  of  y  equal  numerically  but  unlike  in  sign. 
When  :r  =  5,  y  =  0.  When  :r  >  5  the  values  of  y  are  imagi- 
nary ;  this  last  result  means  that  there  is  no  point  with  an 
abscissa  greater  than  5  whose  co-ordinates  will  satisfy  the 
given  equation. 


18 


ANALYTIC    GEOMETRY. 


By  assigning  values  of  x  differing  by  unity,  we  obtain  the 
following  sets  of  values  of  x  and  ?/ ;  and  by  plotting  the  j^^oints, 
and  then  drawing  through  them  a  continuous  curve,  we  obtain 


the  curve  shown  in  Fig.  11. 


Fig.  II. 


dues  of  X. 

Values  of  y 

0     .     .     . 

.      .      zh5. 

1   .   .   . 

.      .      ±4.9. 

9 

.     .     zt4.6. 

3     .     .     . 

.     .     ±4. 

4     .     . 

.     .     ±3. 

5     .     . 

.     .         0. 

-1     .     . 

.     .     ±4.9. 

—  2     . 

.     .     ±4.6. 

-3     .     . 

.     .     ±4. 

--4    .     . 

.     .     ±3. 

-5     .     . 

.     .        0. 

In  this  case,  however,  the  locus  may  be  found  as  follows : 
Let  F  (Fig.  11)  be  any  point  so  placed  that  its  co-ordinates, 
X  =  03f,  y  =  MP,  satisfy  the  equation  x^  -\-y'  =  25.  Join 
OP;  then  a;'  +  y'  =  OP' ;  therefore  0P=  5.  Hence,  if  P  is 
anywhere  in  the  circle  described  with  0  as  centre  and  5  for 
radius,  its  co-ordinates  will  satisfy  the  given  equation  ;  and 
if  P  is  not  in  this  circle,  its  co-ordinates  will  not  satisfy  the 
equation.     This  circle,  then,  is  the  locus  of  the  equation. 

23,  The  points  whose  co-ordinates  satisfy  the  equation 
y2  —  ^x  lie  neither  in  a  straight  line  nor  in  a  circle.  Never- 
theless, they  do  all  lie  in  a  certain  line,  which  is,  therefore, 
completely  determined  by  the  equation.  To  construct  this 
line,  we  first  find  a  number  of  points  which  satisfy  the  equa- 
tion (the  closer  the  points  to  one  another,  the  better)  and  then 
draw,  freehand  or  with  the  aid  of  tracing  curves,  a  continuous 
curve  through  the  points. 

The  co-ordinates  of  a  number  of  such  points  are  given  in 


LOCI    AND    TllElK    EQUATIONS. 


19 


the  table  below.  It  is  evident  that  for  each  positive  value  of 
X  there  are  two  values  of  y,  equal  numerically  but  unlike  in 
sign.  If  we  assume  a  negative  value  for  x,  then  the  value 
of  y  is  imaginary  ;  this  result  means  that  there  are  no 
points  to  the  left  of  the  axis  of  y  which  will  satisfy  the  given 
equation. 

Values  of  a-.  Values  of  y. 

0 0. 

1 ±2. 

2 ±2.83. 

3 ±3.46. 

4 =h4. 

5 ±4.47. 

6 ±4.90. 

7 ±5.29. 

8 ±5.66. 

9 ±6. 

—  1 imaginary. 


Fig.    12. 


In  Fig.  12  the  several  points  obtained  are  plotted,  and  a 
smooth  curve  is  then  drawn  through  them.  It  passes  through 
the  origin,  is  placed  symmetrically  on  both  sides  of  the  axis 
of  X,  lies  w^holly  on  the  right  of  the  axis  of  y,  and  extends 
towards  the  right  without  limit.  It  is  the  locus  of  the  given 
equation,  and  is  a  curve  called  the  Parabola. 

24.  After  a  study  of  the  foregoing  examples,  we  may  lay 
down  the  following  general  principles,  which  form  the  foun- 
dation of  the  science  of  Analytic  Geometry  : 

I.  Every  algebraic  equation  involving  x  and  ?/  is  satisfied 
by  an  unlimited  number  of  sets  of  values  of  x  and  y  ;  in  other 
words,  X  and  y  may  be  treated  as  variables,  or  quantities  vary- 
ing continuously,  yet  always  so  related  that  their  values  con- 
stantly satisfy  the  equation. 


20  ANALYTIC    GEOMETRY. 

II.  The  letters  x  and  y  may  also  be  regarded  as  represent- 
ing the  co-ordinates  of  a  point.  This  point  is  not  fixed  in 
position,  because  x  and  y  are  variables  ;  but  it  cannot  be 
placed  at  random,  because  x  and  y  can  have  only  such  values 
as  will  satisfy  the  equation  ;  now  since  these  values  are  con- 
tinuous, the  point  may  be  conceived  to  Tnove  continuoushj ,  and 
will  therefore  describe  a  definite  line,  or  group  of  lines. 

III.  The  line,  or  group  of  lines,  described  by  a  point 
moving  so  that  its  co-ordinates  always  satisfy  the  ec[uation  is 
called  the  Locus  of  the  Equation ;  conversely,  the  equation  sat- 
isfied by  the  co-ordinates  of  every  point  in  a  certain  line  is 
called  the  Equation  of  the  Line. 

lY.  An  equation,  therefore,  containing  the  variables  x  and 
y  is  the  algebraic  representation  of  a  line ;  it  determines  a 
certain  line  in  the  same  sense  that  two  co-ordinates  determine 
a  certain  point.  And,  conversely,  the  line  (or  group  of  lines) 
which  is  the  locus  of  an  equation  is  the  geometric  representa- 
tion of  the  equation. 

Ex.  6. 

Determine  and  construct  the  loci  of  the  following  equations 
(the  locus  in  each  case  being  either  a  straight  line  or  a  circle)  : 

1.  a:- 6  =  0.  9.  9:r--25  =  0. 

2.  rr  +  5  =  0.  10.  ^x--y-  =  0. 

3.  y  =  -1.  11.  .r^-16/  =  0. 

4.  :r=:0.  12.  .r^  +  /=-36. 

5.  y  =  0.  13.  .T-  +  y--l  =  0. 

6.  x  +  y  =  0.  14.  :r(y+5)  =  0. 

7.  x~2y  =  0.  15.  (^ -  2)  (^-- 3)  =  0. 

8.  2.r-f3y-fl0-0.  16.  (y- 4)  (y  +  1)  =  0. 


LOCI    AND    THEIR    FAiUATlONS.  21 

17.  AVhat    is    llie     gooinetric     meaning    of    the     equation 

Hint.     Resolve  the  equation  into  two  hinoniial  factors, 

18.  What    is    the    geometric    meaning    of    the    equation 

19.  "What    two    lines    form    the    locus    of    the    equation 

20.  Is  the  point  (2,  —  5)  situated  in  the  locus  of  the  equa- 
tion 4:?;-3y  — 22--=:0? 

Hint.     See  if  tlio  co-ordinates  of  the  point  satisfy  tlie  equation. 

21.  Is  the  point  (4,  —6)  in  the  locus  of  the  equation  7/'~9x? 

22.  Is   the  point  (—1,  —1)  in  the  locus  of  the  equation 
16^;^  +  9?/^  -f  15a;  -  Gy  - 18  =  0  ? 

23.  Does  the  locus  of  the  equation  a;^  +  ?/"'^=100  pass  through 
the  point  (—  G,  8)  ? 

24.  Which  of  the  loci  represented  by  the  following  equa- 
tions pass  through  the  origin  ? 

(1)  3.^  +  2  =  0.  (5)  3:^-  =  2y. 

(2)  3x-U7j  +  7  =  0.  (6)  3^'-lly-=0. 

(3)  a;^-lGy2_lO  =  0.  (7)  x' -I67/ =  0. 

(4)  ax  +  h'(/  +  c=--0.  (8)  ax^-hy  =  0. 

25.  The  abscissa  of  a  point  in  the  locus  of  the  equation 
3x  ~  4:7/  —  7  =  0  is  9  ;  what  is  the  value  of  the  ordinate  ? 

Ans.  5. 

26.  Determine  that  point  in  the  locus  of  y^  —  4:r=0  for 
which  the  ordinate  =  -  6.  ^.^^   ^j^^  ^^^-^^  ^.^^  _  ^^^ 

27.  Determine  the  point  where  the  line  represented  by  the 
equation  7x  +  y  -14  —  0  cuts  the  axis  of  x. 

A71S.  The  point  (2,  0). 


22  analytic  geometry. 

Intersections  of  Loci. 

25.  The  term  Curve,  as  used  in  Analytic  Geometry,  means 
any  geometric  locus,  including  the  straight  line  as  well  as 
lines  commonly  called  curves. 

The  Intercepts  of  a  curve  on  the  axes  are  the  distances  from 
the  origin,  measured  along  the  axes,  to  the  points  where  the 
curve  cuts  the  axes. 

26.  To  find  the  intercept  of  a  curve,  having  given  its  equation. 

The  intercept  of  a  curve  on  the  axis  of  x  is  the  abscissa  of 
the  point  where  the  curve  cuts  the  axis  of  x.  The  ordinate  of 
this  point  =  0.  Therefore,  to  find  this  intercept,  put  y  =  0  in 
the  given  equation  of  the  curve,  and  then  solve  the  equation 
for  X  ;  the  resulting  value  of  x  will  be  the  intercept  required. 

If  the  equation  is  of  a  higher  degree  than  the  first  there 
will  be  more  than  one  value  of  x  ;  and  the  curve  will  cut  the 
axis  of  X  in  as  many  points  as  there  are  o^eal  values  of  x. 

If  an  iynaginary  value  of  x  is  found  it  is  to  be  rejected. 
But  in  order  to  make  the  language  of  geometry  correspond  to 
that  of  algebra,  we  may  say  in  this  case  that  the  curve  cuts 
the  axis  of  x  in  an  imaginary  point ;  that  is,  a  point  that  has 
one  or  both  of  its  co-ordinates  imaginary. 

Similarly,  to  find  the  intercepts  on  the  axis  of  y,  put  a-  =  0 
in  the  given  equation,  and  then  solve  it  for  y  ;  the  resulting 
real  values  of  y  will  be  the  intercepts  required. 

27.  To  find  the  points  of  intersection  of  two  curves,  having 
given  their  equations. 

Since  the  points  of  intersection  lie  in  both  curves,  their  co- 
ordinates must  satisfy  both  equations.  Therefore,  to  find  their 
co-ordinates,  solve  the  two  equations,  regarding  the  variables 
X  and  y  as  unknown  quantities. 

If  the  equations  are  both  of  the  first  degree,  there  will  be 


LOCI    AND    THEIR    EQUATIONS.  23 

onl}^  07ie  pair  of  values  of  x  and  y,  and  one  point  of  inter- 
section. 

If  the  equations  are,  one  or  both  of  them,  of  higher  degree 
than  the  first,  there  may  be  several  pairs  of  values  of  x  and 
y ;  in  this  case  there  will  be  as  many  points  of  intersection  as 
there  are  pairs  of  real  values  of  x  and  y. 

If  imaginary  values  of  either  x  or  y  are  obtained,  there  are 
no  corresponding  points  of  intersection. 

28.  If  a  curve  pass  through  the  07'igin,  its  equation,  reduced 
to  its  simplest  for  VI,  cannot  have  a  constant  term ;  that  is,  cannot 
have  a  teo-ni  free  from  both  x  and  y. 

Since  in  this  case  the  point  (0,  0)  is  a  2:»oint  of  the  curve, 
its  equation  must  be  satisfied  by  the  values  x  ^=  0,  and  y  =  0. 
But  it  is  obvious  that  these  values  cannot  satisfy  the  equation 
if,  after  reduction  to  its  simplest  form,  it  still  contains  a  con- 
stant term.  Therefore  the  equation  cannot  have  a  constant 
term. 

29.  If  an  equation  has  no  constant  term,  its  locus  must  pass 
through  the  origiii. 

For,  the  values  x^=0,  y  =  0  must  evidently  satisfy  the  equa- 
tion, and  therefore  the  point  (0,  0)  must  be  a  point  of  the 
locus. 

Ex.  7. 

Find  the  intercepts  of  the  following  curves  : 


1. 

4.1' +  3y- 48  =  0. 

8. 

x-2,  =  0. 

o 

5y  —  3:r  —  30  =^  0. 

9. 

x--9  =  0. 

3. 

x'  +  y'  =  \Q. 

10. 

x'--y-  =  0. 

4. 

9x'-\-^f=U. 

11. 

y'  =  4:x. 

5. 

9.r'-4y^  =  16. 

12. 

x'^y'-^x- 

-  8y  =  32. 

6. 

9x^-4y  =  16. 

13. 

X-  -j-y'^  —  ix  - 

-8y  =  0. 

7. 

a'x'  +  by  =  a'b\ 

14. 

(x-5)'  +  (y- 

~6)=  =  20. 

24  ANALYTIC    GEOMETRY. 

Find  the  j^oints  of  intersection  of  the  following  curves  : 

15.  3a;-4y  +  13  =  0,  ll:r  +  7y  — 104  =  0. 

IG.  2a;  +  3y  =  7,  x  —  y  =  l. 

17.  :r  — 7y+25  =  0,  x'  +  f  =  25. 

18.  3:r  +  4y  =  25,  x'-}-f  =  25. 

19.  ^  +  y  =  8,  :i''  — 3/^  =  34. 

20.  2.r  =  y,  :i;2_|_y2._;L0a7  =  0. 

21.  The  cfjuations  of  the  sides  of  a  triangle  are  2x-j-9y 
+  17=0,  7.y-y-38  =  0,  a:-2y-f  2  =  0.  Find  the  co- 
ordinates of  its  three  vertices. 

22.  The  equations  of  the  sides  of  a  triangle  are  5a7+6y=12, 
3.r  —  4?/  =  30,  ^  +  5y  =  10.     Find  the  lengths  of  its  sides. 

23.  Find  the  lengths  of  the  sides  of  a  triangle  if  the  equa- 
tions of  the  sides  are  x=  0,  'i/  ^=0,  and  4:X  -f-  3y  =  12. 

24.  What  are  the  vertices  of  the  quadrilateral  enclosed  by 
the  straight  lines  x  —  a  =  0,  x  -}-  a  =  0,  y  —  b  =  0,  y  -\~  h  =  0  ? 
What  kind  of  a  quadrilateral  is  it  ? 

25.  Does    the    straight  line   5.r  +  4y  =  20    cut   the    circle 

X'  +  y2  r=  9  ? 

26.  Find  the  length  of  that  part  of  the  straight  line 
3.r  — 4y  =  0  which  is  contained  within  the  circle  :2;^  +  y^  =  25. 

27.  Which  of  the  following  curves  pass  through  the  origin 
of  co-ordinates? 

(1)  7:r-2y  +  4  =  0.  (4)  ax-{-h2j  =  0. 

(2)  7a:-2y  =  0.  (5)  ax +  1?/ -}- c  =  0. 

(3)  f  -  x'  =  4y.  (6)  x'~i-\-a  =  a  +  xy. 

28.  Change  the  equation  4.r  +  2y  — 7=0  so  that  its  locus 
shall  pass  through  the  origin. 


loci  and  their  equations.  25 

Construction  of  Loci. 

30.  If  we  know  that  the  locus  of  a  given  equation  is  a 
straight  line,  the  locus  is  easily  constructed  ;  it  is  only  neces- 
sary to  find  any  two  points  in  it,  plot  them,  and  draw  a 
straight  line  through  them  with  the  aid  of  a  ruler. 

Likewise,  if  we  know  that  the  locus  is  a  circle,  and  can  find 
its  centre  and  its  radius,  the  entire  locus  can  then  be  immedi- 
ately described  with  the  aid  of  a  pair  of  compasses. 

It  will  appear  later  on  that  the  fovTn  of  the  given  equation 
enables  us  at  once  to  tell  whether  its  locus  is  a  straight  line 
or  a  circle. 

If  the  locus  of  an  equation  is  neither  a  straight  line  nor  a 
circle,  then  the  following  method  of  construction,  which  is 
applicable  to  the  locus  of  any  equation  without  regard  to  the 
form  of  the  curve,  is  usually  employed. 

31.  To  construct  the  locus  of  a  given  equation. 

The  steps  of  the  process  are  as  follows  : 

1.  Solve  the  equation  with  respect  to  either  x  or  y. 

2.  Assign  values  to  the  other  variable,  differing  not  much 
from  one  another. 

3.  Find  each  corresponding  value  of  the  first  variable. 

4.  Draw  two  axes,  choose  a  suitable  scale  of  lengths,  and 
plot  the  points  whose  co-ordinates  have  been  obtained. 

5.  Draw  a  continuous  curve  through  these  points. 

Discussion.  An  examination  of  the  equation,  as  shown  in 
the  examples  given  below,  enables  us  to  obtain  a  good  general 
idea  of  the  shape  and  size  of  the  curve,  its  position  with  respect 
to  the  axes,  etc. ;  in  this  way  it  serves  as  an  aid  in  construct- 
ing the  curve,  and  as  a  means  of  detecting  numerical  errors 
made  in  computing  the  co-ordinates  of  the  points.  Such  an 
examination  is  called  a  discussion  of  the  equation. 


26 


ANALYTIC    GEOMETRY. 


Note  1.  This  method  of  constructing  a  locus  is  from  its  nature  an 
approximate  method.  But  the  nearer  the  points  are  to  one  another,  the 
nearer  the  curve  will  approach  the  exact  position  of  the  locus. 

Note  2.  In  theory,  it  is  immaterial  what  scale  of  lengths  is  used. 
In  practice,  the  unit  of  lengths  should  be  determined  by  the  size  of  the 
paper  compared  with  the  greatest  length  to  be  laid  off  upon  it.  Paper 
sold  under  the  name  of  "  co-ordinate  paper,"  ruled  in  small  squares,  j\ 
of  an  inch  long,  on  each  side,  will  be  found  very  convenient  in  practice. 

32.    Construct  the  locus  of  the  equation  ■ 
9.1-2 +  4/ -576  =  0. 
If  we  solve  for  both  x  and  y,  we  obtain  the  following  values  : 

(1) 

(2) 


fV64 


y  

a:  =  ±|Vl44 


r 


By  assigning  to  x  values  differing  by  unity,  and  finding 
corresponding  values  of  ?/,  we  obtain  the  results  given  below. 
To  each  value  of  x,  positive  or  negative,  there  correspond  two 
values  of  ?/,  equal  numerically  and  unlike  in  sign.  By  plotting 
the  corresponding  points,  and  drawing  a  continuous  curve 
through  them,  we  obtain  the  closed  curve  shown  in  Fig.  13. 


Values  of . 

c. 

Values  of  y. 

0    ...     ±12. 

±1 

.     =h  11.91. 

±2 

.     ±  11.62. 

=4=3 

.     ±11.13. 

=±=4 

±  10.39. 

=b5 

.     ±    9.36. 

±6 

±    7.93. 

=h7 

±    5.80. 

±8 

±    0. 

±9 

±  imaginary 

LOCI    AND    TIIEIR   EQUATIONS.  27 

Discussion.  From  equations  (1)  and  (2)  we  see  that  if 
x  =  0,  y  =  ±  12,  and  if  y  =  0,  x  =  ±8;  therefore  the  inter- 
cepts of  the  curve  on  the  axis  of  x  are  +  8  and  —  8,  and  those 
on  the  axis  of  y  are  -f  12  and  —12.  These  intercepts  are  the 
lengths  OA,  OA',  and  OB,  OB',  in  Fig.  13. 

If  we  assign  to  :r  a  numerical  value  greater  than  8,  posi- 
tive or  negative,  we  find  by  substitution  in  equation  (1)  that 
the  corresponding  value  ofy  will  be  imaginary.  This  shows 
that  OA  and  OA'  are  the  maximum  abscissas  of  the  curve. 
Similarly,  equation  (2)  shows  that  the  curve  has  no  points 
with  ordinates  greater  than  -f-12  and  —12. 

The  greater  the  numerical  value  of  x,  between  the  limits 
0  and  +  8  or  0  and  —  8,  the  less  the  corresponding  value  of  y 
numerically  ;  why  ? 

From  equation  (1)  we  see  that  corresponding  to  each 
value  of  X,  between  the  limits  0  and  ±  8,  there  are  tiuo  real 
values  of  y,  equal  numerically  and  unlike  in  sign.  Hence,  for 
each  value  of  x  between  0  and  dr  8  there  are  two  points  of  the 
curve  placed  equally  distan|;  from  the  axis  of  x.  Therefore 
the  curve  is  symmetrical  with  respect  to  the  axis  of  ;r ;  in 
other  words,  if  the  portion  of  the  curve  above  the  axis  of  x 
be  revolved  about  this  axis  through  180°,  it  will  coincide  with 
the  portion  below  the  axis.  Similarly,  it  follows  from  equa- 
tion (2)  that  the  curve  is  also  symmetrical  with  respect  to 
the  axis  of  y.  Therefore  the  entire  curve  is  a  closed  curve, 
consisting  of  four  equal  quadrantal  arcs  symmetrically  placed 
about  the  origin  0.     The  name  of  this  curve  is  the  Ellipse, 

33.    Construct  the  locus  of  the  equation 
4.r-y^  +  16  =  0. 

Solving  for  both  a;  and  y,  we  obtain 

y  =  ±2V^+4,  (1) 

■     x  =  l^:^.  (2) 


28 


ANALYTIC    GEOMETRY. 


We  may  either  assign  values  to  x,  and  then  compute  those 
of  ?/  by  means  of  (1),  or  assign  values  to  y,  and  compute  those 
of  X  by  means  of  (2)  ;  the  second  course  is  better,  because 
there  is  less  labor  in  squaring  a  number  than  in  extracting 
Its  square  root. 

By  assigning  values  to  y,  differing  by  unity  from  0  to  +10, 
and  from  0  to  —10,  and  then  proceeding  exactly  as  in  the 
last  example,  we  obtain  the  series  of  values  given  below,  and 
the  curve  shown  in  Fig.  14. 

Values  of  3/. 

0 
±1 

±2 
±3 

±4 
±5 
±6 
±7 


Fig.   14. 


d=9 
±10 


Values  of  x. 

—  4. 

-3.75. 

-3. 

-1.75. 

0. 

2.25. 

5. 

8.25. 

12. 

16.25. 

21. 

Discussion.  An  examination  of  equations  (1)  and  (2) 
yields  the  following  results,  the  reasons  for  which  are  left 
as  an  exercise  for  the  learner  : 

The  intercepts  on  the  axes  are  : 

On  the  axis  of  x,         OA  =  —  4. 

On  the  axis  of  y,         0B  =  +  4,  and  OC^  -  4. 

If  we  draw  through  A  the  line  AD  1.  to  OX,  the  entire 
curve  lies  to  the  right  of  AD. 

The  curve  is  situated  on  both  sides  of  OX,  and  is  sym- 
metrical with  respect  to  OX. 

The  curve  extends  towards  the  right  without  limit. 


LOCI    AND    THEIR    EQUATIONS. 


29 


The  curve  constantly  recedes  from  OX  as  it  extends 
towards  the  right. 

This  curve  is  called  a  Parabola;  the  point  A  is  called  its 
Vertex,  the  line  ^Xits  Axis. 

34.    Construct  the  locus  of  the  equation 
y  =■  sin  X. 

If  we  assume  for  x  the  values  0°,  10°,  20°,  30°,  etc.,  the  cor- 
responding values  of  y  are  the  natural  sines  of  these  angles, 
and  are  as  follows  : 

Values  of  y. 
0.77. 
0.87. 
0.94. 
0.98. 
1. 

If  we  continue  the  values  of  x  from  90°  to  180°,  the  above 
values  of  y  repeat  themselves  in  the  inverse  order  (e.g.,  if 
X  =  100°,  y  =  0.98,  etc.)  ;  from  180°  to  360°  the  values  of  y 
are  numerically  the  same,  and  occur  in  the  same  order  as 
between  0°  and  180°,  but  are  negative. 


Values  of  x. 

Values  of  ?/. 

Values  of  x.              ^ 

0°     .      .      . 

.      0. 

50°    ...     . 

10°    .     .     . 

.      0.17. 

60°    ...     . 

20°    .     .     . 

.     0.34. 

70°    ...     . 

30°    .     .     . 

.     0.50. 

80°    ...     . 

40°    .     .     . 

.     0.64. 

90°    ...     . 

Fig.   15. 

In  order  to  express  both  x  and  y  in  terms  of  a  common 
linear  unit,  we  ought,  in  strictness,  to  use  the  circular  meas- 
ure of  an  angle  in  which  the  linear  unit  represents  an  angle 


30  ANALYTIC    GEOMETRY. 

of  57.3°,  very  nearly  (see  §  5).  But  it  is  more  convenient, 
and  serves  our  present  purpose  equally  well,  to  assume  that 
an  angle  of  60°  =^  the  linear  unit.  This  assumption  is  made 
in  Fig.  15,  where  the  curve  is  drawn  with  one  centimeter  as 
the  linear  unit. 

Discussion.  The  curve  passes  through  the  origin,  and  cuts 
the  axis  of  .t  at  points  separated  by  intervals  of  180°.  Since 
an  angle  may  have  any  magnitude,  positive  or  negative,  the 
curve  extends  on  both  sides  of  the  origin  without  limit.  The 
maximum  value  of  the  ordinate  is  alternately  +1  and  —1  : 
the  former  value  corresponds  to  the  angle  90°,  and  repeats 
itself  at  intervals  of  360°  ;  the  latter  value  corresponds  to 
the  angle  270°,  and  repeats  itself  at  intervals  of  360°.  The 
curve  has  the  form  of  a  wave,  and  is  called  the  Sinusoid. 

Ex.  8. 
Construct  the  loci  of  the  following  equations  : 

1.  Sx-7j-2  =  0.  13.   y'~l  =  0. 

2.  y  =  2x.  14.   7/  =  x\ 

3.  x'  =  y\%^br,^^iz^UMAj    15.   xij  =  12. 


4. 

x'Jry'  =  '^00. 

16. 

X  =  sin  y. 

5. 

x'-f^25. 

17. 

y  =  2sm  X. 

6. 

^x'-y'  =  0. 

18. 

y  =  sin  2x. 

7. 

^x'  +  9y'  =  U4:. 

19. 

y  =  cos  X. 

8. 

y^-l6x  =  0. 

20. 

y  =  tan  x. 

9. 

y2^16:r  =  0. 

21. 

y  =  cot  X. 

10. 

x'-2x-10y-5-- 

--0. 

22. 

y  =  sec  X. 

11. 

y2_2y_l0a;-=0. 

23. 

y  =  CSC  X, 

12. 

^x-sy+(y-2y= 

=  25. 

24. 

y  =  sin  X  +  cos  x. 

loci  and  their  equations.  31 

Equation  of  a  Curve. 

35.  From  what  precedes,  we  may  conclude  that  every  equa- 
tion involving  x  and  y  as  variables  represents  a  definite  line 
(or  group  of  lines)  known  as  the  locus  of  the  equation. 
Regarded  from  this  point  of  view,  an  equation  is  the  state- 
ment in  algebraic  language  of  a  geometric  condition  which 
must  always  be  satisfied  by  a  point  {x,  y),  as  we  imagine  it 
to  move  in  the  plane  of  the  axes.  For  example,  the  equation 
x  =  2y  states  the  condition  that  the  point  must  so  move  that 
its  abscissa  shall  always  be  equal  to  twice  its  ordinate ;  the 
equation  x^  -\-  y"^  —  A.  states  the  condition  that  the  point  must 
so  move  that  the  sum  of  the  squares  of  its  co-ordinates  shall 
always  be  equal  to  4  ;  etc. 

Conversely,  every  geometric  condition  that  a  point  is 
required  to  satisfy  must  confine  the  point  to  a  definite  line 
as  its  locus,  and  must  lead  to  an  equation  that  is  always 
satisfied  by  the  co-ordinates  of  the  point. 

Hence  arises  a  new  problem,  and  one  usually  of  greater 
difficulty  than  any  thus  far  considered,  namely  : 

Given  the  geometric  condition  to  he  satisfied  by  a  point,  to 
find  the  equation  of  its  locus. 

The  great  importance  of  this  problem  lies  in  the  fact  that 
in  the  practical  applications  of  Analytic  Geometry  the  law  of 
a  moving  point  is  commonly  the  one  thing  known  to  start 
with,  so  that  the  first  step  must  consist  in  finding  the  equa- 
tion of  its  locus. 

Ex.  9. 

1.  A  point  moves  so  that  it  is  always  three  times  as  far  from 
the  axis  of  x  as  from  the  axis  of  y.  AVhat  is  the  equation  of 
its  locus? 

2.  What  is  the  equation  of  the  locus  of  a  point  which  moves 
so  that  its  abscissa  is  always  equal  to-f6?     — G?     0? 


32  ANALYTIC    GEOMETRY. 

3.  What  is  the  equation  of  the  locus  of  a  point  which  moves 
so  that  its  ordinate  is  always  equal  to +4?     — 1?     0? 

4.  A  point  so  moves  that  its  distance  from  the  straight  line 
:r  =  3  is  always  numerically  equal  to  2.  What  is  the  equation 
of  its  locus  ? 

5.  A  point  so  moves  that  its  distance  from  the  straight  line 
y  =  6  is  always  numerically  equal  to  3.  Find  the  equation 
of  its  locus.     Construct  the  locus. 

6.  A  point  moves  so  that  its  distance  from  the  straight 
line  :^  -f  4  =  0  is  always  numerically  equal  to  5.  Find  the 
equation  of  its  locus.     Construct  the  locus. 

7.  What  is  the  equation  of  the  locus  of  a  point  equidistant 

(1)  from  the  parallels  x  =  0  and  a;  =  —  6  ? 

(2)  from  the  parallels  y  =  7  and  ?/  —  —  3  ? 

8.  What  is  the  equation  of  the  locus  of  a  point  always 
equidistant  from  the  origin  and  the  point  (6,  0)  ? 

Find  the  equation  of  the  locus  of  a  point 
9.    Equidistant  from  the  points  (4,  0)  and  (—  2,  0)". 

10.  Equidistant  from  the  points  (0,  —  5)  and  (0,  9). 

11.  Equidistant  from  the  points  (3,  4)  and  (5,  —2). 

12.  Equidistant  from  the  points  (5,  0)  and  (0,  5). 

13.  A  point  moves  so  that  its  distance  from  the  origin  is 
always  equal  to  10.     Find  the  equation  of  its  locus. 

14.  A  point  moves  so  that  its  distance  from  the  point  (4,  —  3) 
is  always  equal  to  5.  Find  the  equation  of  its  locus,  and  con- 
struct it.  What  kind  of  curve  is  it  ?  Does  it  pass  through 
the  origin  ?     Why  ? 

15.  What  is  the  equation  of  the  locus  of  a  point  whose 
distance  from  the  point  (—  4,  —  7)  is  always  equal  to  8  ? 


LOCI    AND    THEIR    EQUATIONS.  33 

16.  About  the  origin  of  co-ordinates  as  centre,  with  a  radius 
equal  to  5,  a  circle  is  described.  A  point  outside  this  circle 
so  moves  that  its  distance  from  the  circumference  of  the  circle 
is  always  equal  to  4.     What  is  the  equation  of  its  locus  ? 

17.  A  high  rock  A,  rising  out  of  the  water,  is  3  miles 
from  a  perfectly  straight  shore  BC.  A  vessel  so  moves  that 
its  distance  from  the  rock  is  always  the  same  as  its  distance 
from  the  shore.     What  is  the  equation  of  its  locus  ? 

18.  A  point  A  is  situated  at  the  distance  6  from  the  line 
J^C,  A  moving  point  P  is  always  equidistant  from  A  and 
BC.     Find  the  equation  of  its  locus. 

19.  A  point  moves  so  that  its  distance  from  the  axis  of  x  is 
half  its  distance  from  the  origin  ;  find  the  equation  of  its  locus. 

20.  A  point  moves  so  that  the  sum  of  the  squares  of  its 
distances  from  the  two  fixed  points  (a,  0)  and  (—a,  0)  is  the 
constant  2^' ;  find  the  equation  of  its  locus. 

21.  A  j)oint  moves  so  that  the  difference  of  the  squares  of 
its  distances  from  (a,  0)  and  (—  a,  0)  is  the  constant  k^ ;  find 
the  equation  of  its  locus. 

Ex.  10.      (Review.) 

1.  If  we  should  plot  all  possible  points  for  which  x  =  —  5, 
how  would  they  be  situated  ? 

2.  Construct  the  point  (x,  y)  if  a:=  2  and 

(1)  y  =  4^'-3,     (2)  3a; -2?/ =  8. 

3.  The  vertices  of  a  rectangle  are  the  points  («,  h),  (—a,  Z>), 
(—a,  —  ^),  and  (a,  —  i).  Find  the  lengths  of  its  sides,  the 
lengths  of  its  diagonals,  and  show  that  the  vertices  are  equi- 
distant from  the  origin. 

4.  What  does  equation  [1],  p.  6,  for  the  distance  between 
two  points,  become  when  one  of  the  points  is  the  origin  ? 


34  ANALYTIC    GEOMETRY.  > 

5.  Express  by  an  equation  that  the  distance  of  the  point 
(x,  y)  IVoui  the  point  (4,  G)  is  equal  to  8. 

6.  Express  that  the  point  {x,  y)  is  equidistant  from  the 
points  (2,  3)  and  (4,  5). 

7.  Find  the  point  equidistant  from  the  points  (2,  3),  (4,  5), 
and  (6,  1).     What  is  the  common  distance  ? 

8.  Prove  that  the  diagonals  of  a  rectangle  are  equal. 

9.  Prove  that  the  diagonals  of  a  parallelogram  mutually 
bisect  each  other. 

10.  The  co-ordinates  of  three  vertices  of  a  parallelogram 
are  known :  (5,  3),  (7,  10),  (13,  9).  What  are  the  co-ordi- 
nates of  the  remaining  vertex  ? 

11.  The  co-ordinates  of  the  vertices  of  a  triangle  are  (3,  5), 
(7,  —9),  (2,  —4).  Find  the  co-ordinates  of  the  middle  points 
of  its  sides. 

12.  The  centre  of  gravity  of  a  triangle  is  situated  on  the 
line  joining  any  vertex  to  the  middle  point  of  the  opposite 
side,  at  the  point  of  trisection  nearest  that  side.  Find  the 
centre  of  gravity  of  the  triangle  whose  vertices  are  the  points 
(2,3),(4,-5),  (-3, -6). 

13.  The  vertices  of  a  triangle  are  (5,  -  3),  (7,  9),  (-  9,  6). 
Find  the  distance  from  its  centre  of  gravity  to  the  origin. 

14.  The  vertices  of  a  quadrilateral  are  (0,  0),  (5,  0),  (9,  11), 
(0,  3).  Find  the  co-ordinates  of  the  intersection  of  the  two 
straight  lines  which  join  the  middle  points  of  its  opposite  sides. 

15.  Prove  that  the  two  straight  lines  which  join  the  middle 
points  of  the  opposite  sides  of  any  quadrilateral  mutually 
bisect  each  other. 

16.  A  line  is  divided  into  three  equal  parts.  One  end  of 
the  line  is  the  point  (3,  8) ;  the  adjacent  point  of  division  is 
(4,  13).     What  are  the  co-ordinates  of  the  other  end? 


LOCI    AND    THEIR    EQUATIONS.  35 

17.  The  line  joining  the  jDoints  (^'i,  7/1)  and  (x.^,  7/2)  is  divided 
into  four  equal  parts.  Find  the  co-ordinates  of  the  jooints  of 
division. 

18.  Explain  and  illustrate  the  relation  which  exists  between 
an  equation  and  its  locus. 

19.  Construct  the  two  lines  which  form  the  locus  of  the 
equation  x'^  —  7x  =  0. 

20.  Is  the  point  (2,   —  5)    in    the   locus   of  the    equation 

21.  The  ordinate  of  a  certain  point  in  the  locus  of  the 
equation  :i-- +  /  +  20a;  —  70  =  0  is  1.  What  is  the  abscissa 
of  this  point  ? 

22.  Find  the  intercepts  of  the  curve  x^-\-y  —  5x—7y-{-6  =  0. 

Find  the  points  common  to  the  curves  : 

23.  x'  +  7/  =  100,  and  7/  — 12 x  =  0. 

24.  x'^  +  y'  =  5a^  and  rc^  —  4  ay. 

25.  h'-x"- -{- ahf  ^  d'b\  and  x-  +  y-'  =  a\ 

26.  Find  the  lengths  of  the  sides  of  a  triangle,  if  its  ver- 
tices are  (6,  0),  (0,  -8),  (-4,  -2). 

27.  A  point  moves  so  that  it  is  always  six  times  as  far  from 
one  of  two  fixed  perpendicular  lines  as  from  the  other.  Find 
the  equation  of  its  locus. 

28.  A  point  so  moves  that  its  distance  from  the  fixed 
point  A  is  always  double  its  distance  from  the  fixed  line  AB. 
Find  the  equation  of  its  locus. 

29.  A  fixed  point  is  at  the  distance  a  from  a  fixed  straight 
line.  A  point  so  moves  that  its  distance  from  the  fixed  point 
is  always  twice  its  distance  from  the  fixed  line.  Find  the 
equation  of  its  locus. 


CHAPTER  11. 

THE    STRAIGHT    LINE. 

Equations  of  the  Straight  Line. 

36.  Notation.  Throughout  this  chapter,  and  generally  in 
equations  of  straight  lines, 

a  =  the  intercept  on  the  axis  of  x.  / 

h  =  the  intercept  on  the  axis  of  y.  ^ 

y  =  the  angle  between  the  line  and  the  axis  of  x. 
971  =  tan  y.    ' 
p  =  the  distance  of  the  line  from  the  origin. ' 

a  =  the  angle  between  p  and  the  axis  of  x.  "^ 

These  six  quantities  are  constants  for  a  given  straight  line, 
but  vary  in  value  for  different  lines  :  a,  b,  m,  and  p  may 
have  any  values  from  -co  to  -}-co;  y  and  a,  any  values  from 
0°  to  360°. 

The  constant  m  is  often  called  the  Slope  of  the  line ;  its 
value  determines  the  direction  of  the  line. 

In  order  to  determine  a  straight  line,  two  geometric  con- 
ditions must  be  given. 

37.  To  find  the  equation  of  a  straight  line  passing  through 
two  given  points  {x^,  y^  and  {x2,  yi). 

Let  A  (Fig.  16)  be  the  point  {x^,  y^),  B  the  point  {x.^,  y^)  ; 
and  let  P  be  any  other  point  of  the  line  drawn  through  A 
and  jB,  X  and  y  its  co-ordinates.  Draw  AC,  J5D,  PIT,  J.  to 
OX  and  ^^i^  II  to  OX. 


THE    STRAIGHT    LINE. 


37 


The  triangles  APF,  ABE  are  similar  ;  therefore 

PF  _^BE 
AF     AE 

'No^y  FF=  7/  — 2/i,  AF^-x  —  Xi,  BE=y.j_—y^,  AE^x^—x^. 


Therefore 


1  ^2  •*'l 


This  is  the  equation  required. 


[4] 


Y 

F 

i          \D 

M 

^              ^ 

0      \        X 

Fig.  16. 


Fig.  17. 


It  is  evident  that  the  angle  PAF=y.  Therefore  each 
side  of  equation  [4]  is  equal  to  tan  y  or  m.  The  first  side 
CO  ntains  the  two  variables  x  and  y,  and  the  equation  tells  us 

th  at  they  must  vary  in  such  a  way  that  the  fraction  ^ — ^ 
(  .  _  X  —  x^ 

shall  remain  constant  in  value,  and  always  equal  to  m. 

Note.  In  Fig.  16  the  points  A,  B,  and  P  are  assumed  in  the  first 
quadrant  in  order  to  avoid  negative  quantities.  But  the  reasoning  will 
le  ad  to  equation  [4]  whatever  be  the  positions  of  these  points.  In 
Fis.  17  the  points  are  in  different  quadrants.  The  triangles  APF, 
/\BE  are  to  be  constructed  as  shown  in  the  figure.  They  are 
similar ;  and  by  taking  proper  account  of  the  algebraic  signs  of  the 
q  uantities,  we  arrive  at  equation  [4],  as  before.  The  learner  should 
sjtudy  this  case  with  care,  and  should  study  other  cases  devised  by  him- 
stdf,  till  he  is  convinced  that  equation  [4]  is  perfectly  general. 

i 


38 


ANALYTIC    GEOMETRY. 


38.    To  find  the  equation  of  a  straight  line,  given  one  point 
(^^\j  1/y)  ?^  ^^^  ^^^^'<^  <^^-^  i^^(^  angle  y. 

Let  the  figure  be   constructed   like   Fig.   16,  omitting  the 
point  B  and  the  line  BED.     Then  it  is  evident  that 


whence, 


AF      X  -  X, ' 
2/  —  2/i  =  m{oc  —  Xj), 


Y 

B 

y^... _Jn 

AX 

y     0 

U    X 

[5] 


Fis.   18. 


39.  To  find  the  equation  of  a  straight  li^ie,  given  the  inteTcc]^)^^ 
h  and  the  angle  y. 

Let  the  line  cut  the  axes  in  the  points  A  and  Jj  (Fig.  LS). 
Let  F  be  any  point  (x,  y)  in  the  line.  Draw  FM 1.  to  0\X, 
and  EC  II  to  OX. 

Then  OE^h,  FEC=y,  EC=x,  FC=y-h]  i 


n 


40.    To  find  the  equation  of  a  straight  line,  given  'its  intercept 's 
a  and  b.  ^ 

Let  the  line  cut  the  axes  in  the  points^  and  E  (Fig.  19  c, 
and  let  F  be  any  other  point  {x,  y)  in  the  line.    Then  OA  =  ^  i, 


therefore 

.7        ^ 

'''            X       ' 

whence 

y  =  inx  -\-  h. 

THE    STRAIGHT    LINE. 


39 


OB  =  h.     Draw  7\1/_L  to  OX.     The  triangles  FMA,  BOA 
are  similar  ;  therefore 

PM     MA      OA-OM 


BO       OA 

OA 

r-«---l_ 

X 

h          a 

a' 

or 


whence  -  +  ,,  =1.  [7] 

^  This  is  called  the  Symmetrical  Equation  of  the  straight  line. 

41.    To  find  the  cquatio7i  of  a  straight  line,  given  its  distance 
p  from  the  origin,  and  the  angle  a. 


Y 

B 

s 

/ 

\^     . 

0 

M 

Vx 

Fig.  20. 

Let  AB  (Fig.  20)  be  the  line,  F  any  point  in  it.  Draw 
OS  ±  to  AB,  meeting  AB  in  S ;  PM  ±  to  OX;  MR  II  to 
AB,  meeting  08  in  R  ;  and  PQ  ±  to  AB. 

Then  p  =  OS  =0P+  QP,  a  =  SOM=  PMQ. 

By  Trigonometry 

OP  =  Oil/ cos  a  =  X  cos  a, 
QP  ~  PM  sin  a  =  y  sin  a. 
Therefore    •  Oi?4- (?P  =  ;7  =  a:  cos  a +  ?/ sin  a. 

Or  a?  cos  a  -I-  2/  sin  a  =  p.  rg] 

This  is  called  the  Normal  Equation  of  the  straight  line. 
The  quantity  p  is  ahvags  j^ositive  (like  the  radius  of  unit 
circle  in  Trigonometry). 


40 


ANALYTIC    GEOMETRY. 


Note  1.     Observe  that  all  the  equations  of  the  straight  line  which 
have  been  obtained  are  of  the  first  degree. 

Note  2.     For  the  value  of  the  sines,  cosines,  and  tangents  of  the  more 
common  angles,  see  Appendix. 

Ex.  11. 

Find  the  equation  of  the  straight  line  passing  through  the 
two  points : 

1.  (2,  3)  and  (4,  5). 

2.  (4,  5)  and  (7,  11). 

3.  (-1,2)  and  (3,-2). 

4.  (-2,-2)  and  (-3,-3). 

5.  (4,  0)  and  (2,  3). 

6.  (0,  2)  and  (-3,0). 


7. 

(2,  5)  and  (0,  7). 

8. 

(3,  4)  and  (0,  0). 

9. 

(3,  0)  and  (0,  0). 

10. 

(3,  4)  and  (-2,4). 

11. 

(2,  5)  and  (-2, -5). 

12. 

(??i,  ?2)  and  (-???,  —  7i). 

Find  the  equation  of  a  straight 

13.  (4,  1)  andy-r45°. 

14.  (2,  7)  andy  =  60°. 
(1^  (-3,  11)  and  y  =  45°. 

16.  (13,-4)  and  7  =  150°. 

(ft.  (3,  0)  and  7  =  30°. 

18.  (0,  3)  and  7  =  135°. 

19.  (0,  0)  and  7  =  120°. 

20.  (2,-3)  and  7  =  0°. 

21.  (2, -3)  and  7=90°. 
(^^  i.=  2and7  =  45°. 

23.  ^  =  5,7=45°. 

24.  Z»  =  -4,  7  =  45°. 

25.  5^-4,7  =  30°. 
2^  ^=:_4,  7  =  0°. 

27.  5  =  -4,  7  =  60°. 

28.  5  =  -4,  7  =  90°. 


line,  given  : 

$3':    ^  =  -4,7=120°. 

30.  b  =  -i,  7  =  135°. 

31.  ^>  =  - 4,  7  =  150°. 
(B^   ^  =  -4,7  =  225°. 

33.  a  =  4,  6  =  3. 

34.  a  =  -6,b  =  2. 

35.  a^=  — 3,  6  =  —  3. 
^   a  =  c>,  5  =  -3. 

37.  a  =  -10,  h  =  D. 

38.  a  =  l,  b=-l. 
(39y  a^n,  h  =  —  n. 

40.  a  =  ?2,  h^in. 

41.  p  =  5,  a  =  45°. 
42j  p=--^,  a  =  120°. 
43"^  ;^  =  5,  a=240°. 
44.   »  =  5,  a  =  300°. 


THE    STRAIGHT    LINE.  41 

Write  the  equations  of  the  sides  of  a  triangle  : 

45.    If  its  vertices  are  the  points  (2,  1),  (3,  —  2),  (-4,  -1). 

4G.    If  its  vertices  are  the  points  (2,  3),  (4,  —  5),  (-  3,  -  G). 

47.  Form  the  equations  of  the  medians  of  the  triangle 
described  in  No.  53. 

48.  The  vertices  of  a  quadrilateral  are  (0,  0),  (1,  5),  (7,  0), 
(4,  —  9).  Form  the  equations  of  its  sides,  and  also  of  its 
diagonals. 

Find  the  equation  of  a  straight  line,  given  : 

49.  a  =  7i,  7-30°.  51.  p  =  6,  y  =  45°. 

50.  a  =  -  3,  (x„  y,)  =  (2,  5).      52.  ;?  ^  6,  y  -  225^ 

Note.  The  best  way,  in  general,  to  construct  a  straight  line  from  its 
equation  is  to  find  its  intercepts  (f  26),  and  then  lay  them  off  on  the 
axes.  If  the  line  passes  through  the  origin,  it  has  no  intercepts ;  but 
in  this  case  a  second  point  in  the  line  is  easily  found  by  assuming  any 
convenient  value  (as  1)  for  x,  and  then  computing  the  corresponding 
value  of  7/  from  the  given  equation. 

The  intersection  of  two  straight  lines  is  to  be  found  by  the  general 
method  explained  in  ^  27. 

Construct  the  following  lines,  and  find  the  point  of  inter- 
section : 

53.  3.r  — 2y  +  ll=0  and  ?/  =  7rr. 

54.  3a'  +  5y-13  =  0  and  4:i--?/-2  =  0. 

55.  2a: +  3?/ =  7  and  x  —  y  =  \. 

56.  y-f  13  =  5a;  and  y +  19  =1  x. 

57.  3.2;  +  y+12  =  0  and  2y=3a;-6. 

58.  1  +  1  =  1  and  1  +  1  =  0. 

59.  Find  the  vertices  of  the  triangle  whose  sides  are  the 
lines  2a: +  9?/ +13  =  0,  y  =  7a:-38,  2y-a:=:2. 


42  ANALYTIC    GEOMETRY. 

60.  Find  the  equation  of  the  straight  Hne  passing  through 
the  origin  and  the  intersection  of  the  lines  3a;  —  2y  +  4  =  0 
and  3:r  -|-  43/  —  5.  Also  find  the  distance  between  these  two 
points. 

61.  What  is  the  equation  of  the  line  passing  through  (xi,  3/1), 
and  equally  inclined  to  the  two  axes  ? 

62.  Find  the  equations  of  the  diagonals  of  the  parallelo- 
gram formed  by  the  lines  x  =  a,  x  =  b,  x  —  c,  x^=d. 

63.  Show  that  the  lines  ?/  =  2:r  +  3,  ?/  =  3:r  +  4,  y  =  4.r  +  5 
all  pass  through  one  point. 

Find  the  intersection  of  two  of  the  lines,  and  then  see  if  its  co-ordi- 
nates will  satisfy  the  equation  of  the  remaining  line. 

64.  The  vertices  of  a  triangle  are  (0,  0),  (x^,  0),  (x^,  y.^. 
Find  the  equations  of  its  medians,  and  prove  that  they  meet 
in  one  point. 

65.  What  must  be  the  value  of  m  if  the  line  y  =  mx  passes 
through  the  point  (1,  4)? 

66.  The  line  y  =  mx-{-2>  passes  through  the  intersection  of 
the  lines  y  =  x-{-l  and  y  ==  2:zr  +  2.    Determine  the  value  of  vi. 

67.  Find  the  value  of  h  if  the  line  y  =  ^x-\-h  passes  through 

the  point  (2,  3). 

68.  What  condition  must  be  satisfied  if  the  points  (.ri,  yi), 
(^2,  y-i),  (^3,  y-i)  lie  in  one  straight  line  ? 

Hint.  Let  equation  [4]  represent  the  line  through  {x^,  y{)  and 
(xj,  2/2) ;    ^^^^  (■^3'  2/3)  inust  satisfy  it. 

69.  Discuss  equation  [5]  for  the  following  cases  :  (i.)  (:i'i,  y^) 
=  (0,  0),  (ii.)m=-0,  (iii.)m  =  oo. 

70.  Discuss  equation  [6]  for  the  following  cases  :  (i.)  b  =  0, 
(ii.)m  — 0,  (iii.)  ?7i  =  CO,  (iv.)  ??^  ~  0,  and  5  =  0. 

71.  Discuss  equation  [7]  for  the  following  cases  :  (i.)  a  =  b, 
(ii.)a  =  0,  (iii.)a— 00,  (iv.)6  =  oo. 


the  straight  line.  43 

General  Equation  of  the  First  Degree. 

42.  Every  equation  involving  x  and  y  as  variables,  which 
can  be  reduced  by  algebraic  operations  to  the  form 

y^mx^  b, 
represents  a  straight  line  having  for  slope  the  coefficient  of  x, 
and  for  interccp)t  on  the  axis  of  y  the  value  of  the  constant  term. 

Algebraic  operations  never  change  the  values  of  x  and  y 
which  will  satisfy  an  equation  ;  therefore  they  cannot  change 
the  locus  represented  by  the  equation. 

The  equation  y  =  mx-\-b,  from  the  manner  in  which  it  was 
established,  necessarily  holds  true  for  all  values  of  m  and  b 
from  —00  to  +00.  Therefore  every  equation  of  the  form 
y^^mx-\-b  must  represent  some  particular  one  of  the  infi- 
nite number  of  straight  lines  obtained  by  giving  all  pos- 
sible values  to  the  general  symbols  on  and  b  ;  and  the  values 
of  the  two  constants  in  the  equation  must  be  the  values  of 
VI  and  b  for  the  particular  line  represented  by  the  equation. 

43.  Every  equation  which  can  be  put  in  the  form 

a      0 
represents  a  straight  line,  having  for  its  intercepts  the  denomi- 
nators of  X  and  y,  respectively. 

The  proof  is  similar  to  that  of  the  preceding  proposition. 

44.  Every  equation  of  the  first  degree,  with  respect  to  the 
variables  x  and  y,  can  be  put  in  the  form 

Ax  +  By  +  C  =  0.  [9] 

where  A,  B,  C  stand  for  any  numbers,  positive  or  negative, 
entire  or  fractional,  rational  or  irrational.  A  and  B  cannot, 
however,  both  be  0  ;  for  if  they  could,  then  we  should  also  have 
C*=0,  and  the  equation  would  vanish  entirely.  Equation  [9] 
is  termed  the  General  Equation  of  the  First  Degree. 


44  ANALYTIC    GEOMETKY. 

45.  Every  equation  of  the  first  degree  in  x  and  y  represents  a 
straiyht  line. 

If  we  reduce  the  equation  to  the  form  Ax  -}-  By  -f  C=  0, 

and  solve  for  y,  we  obtain 

A         C 

y=-B'-B 

But  the  equation  has  now  the  form  y  =  mx  +  b  ;  therefore 
it  represents  a  straight  line  (§  42),  and  the  values  of  the  slope 
and  the  intercept  on  the  axis  of  y  are 

A  .  C 

m-= .         0  =  — -• 

B  B 

46.  To  find  the  intercepts  and  slope  of  a  straight  line,  having 
given  its  equation. 

Method  I.    Find  the  intercepts,  a  and  h,  as  in  §  26.     Then 

(Fig.  19)  m  =  tan7-=-tan(180°-y)=--tan^^O-=-- 

a 

Method  II.  Reduce  the  given  equation  to  the  form  [6] ; 
and  make  m  equal  to  the  coefficient  of  x.  Reduce  the  equa- 
tion to  the  form  [7],  and  make  a  and  b  respectively  equal  to 
the  denominators  of  x  and  y.  For  these  operations  amount 
to  nothing  more  than  substituting  particular  values  in  the 
place  of  general  symbols  which  from  their  nature  include  all 
assignable  values  whatsoever. 

Note.  This  mode  of  obtaining  from  the  equation  of  a  curve  the  vahies 
of  its  constants  is  sometimes  called  the  Method  of  Equating  Coefficients. 

Method  III.  Deterruine  the  values  of  ???,  a,  and  h,  in  terms 
of  ^,  B,  and  0,  by  the  method  of  equating  coefficients.  The 
results  are  A  C  C 

™=__.  .=--,  b  =  --- 

Nothing  now  remains  but  to  reduce  the  given  equation  to  the 
general  form,  Ax^By  +  C=  0,  and  then  substitute  the  par- 
ticular values  of  ^,  ^,  and  C. 


/ 

THE    STRAIGHT    LINE.  45 

47.    Find  the  intercepts  and  slope  of  the  straight  line  repre- 
sented by  the  equation  3  a:  — 3  =  4y  +  9. 

By  I.    If  y  =  0,  a;  =  4  ;    if  a;  =  0,  y  =  -  3. 

Therefore         ci  —  y,  h  =  —  Z,  m  = = .^  =  7- 

a  4        4 

By  II.    The  equation  put  into  the  forms  [6]  and  [7]  becomes 

y  =  |,r-3,  and|  +  -^=l. 

3 
Therefore         a  =  4,   ^  —  —  3,    m  =  -- 

4 

By  III.    The  given  equation  in  the  form  [9]  is 

3:r-4y-12  =  0. 

Therefore         A  =  2>,  ^  =  -4,  C=-12,  and 

-12       .     ;  -12  Q  3        3 

a  =  —^  =  ^,   b  =  -—=-S,  r.  =  -  —  =  -. 

Ex.  12. 

Describe  the  position  of  the  following  lines  by  determining 
the  values  of  a,  b,  and  m. 

1.  1  +  ^  =  1.  10.  1-1  =  1. 

2.  y  =  ^-9.  11.  |  +  |  =  -l- 

3.  3a:  +  2  =  2y  12.  3y  =  rr. 

4.  4y  =  5a:.  13.  3rr  =  y. 

5.  7a:  +  3y  =  0.  14.  5.r-4y  +  20  =  0. 

6.  4y  =  3:^  +  24.  15.  y  =  6.r+12. 

7.  x  +  i/  =  S.  16.  y  +  2-:a;-4. 

8.  4y  +  :r  +  ll=0.  17.  a:  +  V3?/-fl0  =  0. 

9.  5:c-3y  +  15  =  0.  18.  :r— V3y-10  =  0. 


4G  ANALYTIC    GEOMETE-Y. 

19.  Discuss  equation  [9]  for  the  following  cases  : 

(i.)  ^  =  0.     (iv.)  A  =  c^.  (vii.)  A=B,  0=0. 

(ii.)  ^  =  0.      (v.)  A  =  C=0.  (viii.)  A  =  -B,  C=Q. 
(iii.)  C  =  0.     (vi.)  A  =  B. 

20.  Eeduce  equation  [7]  to  the  form  of  equation  [6],  and 
find  the  value  of  m  in  terms  of  a  and  b. 

21.  "What  value  must  C  have  in  order  that  the  line 
4iX  —  5i/-\-  C  may  pass  through  the  origin?  Through  the 
point  (2,  0)  ? 

22.  Determine  the  values  of  ^,  B,  and  C,  so  that  the  line 
Ax -\- By -j- C  =^  0  may  pass  through  the  points  (3,0)  and 
(0,-12). 

Since  the  co-ordinates  of  the  given  points  must  satisfy  the  equation, 
we  have  the  two  relations  oA  -^  C  =  0  and  — 12^  -f  C=  0. 

23.  From  equation  [9]  deduce  equation  [4]  by  the  method 
used  in  solving  No.  22. 

24.  If  equations  [4]  and  [9]  represent  the  same  line,  what 
are  the  values  A,  B,  C,  in  terms  of  Xi,  t/^,  Xo  and  3/2  ? 

25.  In  equation  [4]  find  the  values  of  m  and  h  in  terms  of 

^X,    2/1,    ^2,   2/2. 

26.  In  equation  [8]  find  the  values  of  cos  a,  sin  a,  and  ^,  in 
terms  of  the  general  constants  A,  B,  C. 

To  reduce  the  form  [9]  to  the  form  [S]  we  must  multiply  each  term 
by  a  certain  quantity  Ic ;  to  determine  h,  we  have 

lii^Ax  —  By  -\-  C)  =  X  cos  o  +  ysino— jo  =  0, 
whence,  by  equating  coefficients,  we  obtain 

cos  a  =  kA,    sin  o  =  hB,  p  =  —  hC. 
But                        cos^  a  +  sm^a  =  1, 
whence  we  have     k=  ± 


\/A'  +  B' 
By  substituting  the  value  of  k,  we  have 

A*  ^.  B  C 


y/A}  +  ^2  ±  ^^2  ^^i  ±  ^j^2  ^  ^2 


THE   STRAIGHT    LINE. 


47 


and  we  should  choose  that  sign  before  the  radical  which  will  make  p 
positive  (^  41),  therefore  the  sign  which  is  unlike  that  of  C. 

27.  What  are  the  values  of  cos  a,  sin  a,  and  j9  for  the  fol- 
lowing lines?  Construct  the  lines.  Show  that  they  enclose  a 
parallelogram. 

(a)  12a: +  52/ -26-0.        (c)    12.r- Sy- 26  =  0. 

(b)  12:c  +  5y  +  26  =  0.        (d)  12:^- -  5y  +  26  =  0. 


Parallels  and  Perpendiculars. 

48.  If  the  lines  represented  by  the  equations  y^7nx-\-b 
and  y  =  7n'x-{-  h'  are  parallel,  then  we  must  have,  by  Geom- 
etry, 


and  therefore 


y; 

VI, 


or 


Fig.   21. 

If  the  two  lines  are  perpendicular,  then,  by  Geometry, 

y  =  y  +  90°  (see  Fig.  21). 

Therefore,  by  Trigonometry, 

tan  y'  =  —  cot  y, 

1 

m'  = . 

7)1 


or 


mm' 


1. 


48  ANALYTIC    GEOMETRY. 

49.  Conversely,  prove  that  the  lines  represented  hy  the  equa- 
tions y  =  inx  +  Z>  and  y  =^  on^x  +  h'  are  parallel  if  on^^m' ; 
perpendicular,  if  ??^.??^'  =  —  1. 

Hint.     Use  the  above  reasoning,  taken  in  reverse  order. 

Not::.  The  equations  m  =  m'  and  mm'  =  —  1  are  examples  of  Equa- 
tions of  Condition ;  they  express  tlve  conditions  which  must  be  satisfied 
in  order  that  two  lines  may  be  parallel  or  perpendicular,  respectively. 

50.  To  find  the  equations  of  a  straight  line  passing  through 
the  point  (.r^,  ?/i)  and  (i.)  parallel,  (ii.)  perpendicular,  to  the 
line  y  =  r}ix  +  h. 

The  slope  of  the  required  line  is  on  in  case  (i.),  and 

in  case  (ii.) ;  and  in  both  cases  the  line  passes  through  a  given 
point  (rri,  3/i). 

Therefore  (§  38)  the  required  equations  are 

(1)  y  —  yi  =  'in{x  —  x;), 

(2)  2/-yi  =  --(^-^i). 

51.  We  shall  now  solve  the  problem  of  §  50  in  another  way. 
Let  the  equation  of  a  given  line  be 

Ax  +  By  +  C-=^0.  (1) 

If  we  change  C  to  any  other  value  K,  hut  leave  A  and  B 
unchanged,  the  new  equation 

Ax^By^K=^  (2) 

will  represent  a  line  parallel  to  the  given  line,  because  the 
two  lines  have  the  same  slope  (§  49). 

If  we  change  C  to  K,  as  before,  and  also  interchange  the 
coefficients  A  and  B,  and  alter  the  sign  of  one  of  them,  the  new 
equation  Bx-Ay  +  K=Q  (3) 

will  represent  a  line  perpendicular  to  the  given  line,  because 
the  slopes  of  the  two  lines  satisfy  the  condition  ?7im'=  — 1. 


THE    STRAIGHT    LINE.  49 

By  assigning  different  values  to  K,  equations  (1)  and  (2) 
may  be  made  to  represent  different  parallels,  and  different 
perpendiculars,  respectively,  to  the  given  line  ;  and  if  we 
regard  K  as  entirely  undetermined,  then  equation  (2)  may 
be  said  to  represent  all  parallels,  and  equation  (3)  all  per- 
pendiculars, to  the  given  line. 

But  if  in  either  case  we  assign  a  particular  value  to  II,  or 
add  a  new  condition  which  determines  A'',  then  the  equation 
will  represent  one  definite  straight  line. 

Suppose  we  add  the  condition  that  the  line  must  pass 
through  a  given  point  {xi,  y^  ;  then  these  co-ordinates  must 
satisfy  equations  (2)  and  (3),  and  we  have 

Ax^-{-Bij^-^K=0,  and  Bx^-Ay^-^K=0. 
Hence      K=^  —  {Ax^  -\-  By^),  and  'K=^  Ay^  —  Bxi. 
Substituting  these  values  of  K  in  (2)  and  (3),  we  obtain  for 
the  equations  of  the  required  lines 

Ax-^By  =  Ax,+B7/,.  (4) 

Bx  —  Ay  =  Bxi  —  Ayi.  (6) 

Ex.  13. 
Find  the  equation  of  a  straight  line 

1.  Passing  through  (3,  —7),  and  II  to  the  line  y=Sx  —  b. 

2.  Passing  through  (5,  3),  and  II  to  the' line  ^y~\x  =  l. 

3.  Passing  through  (0,  0),  and  il  to  the  line  y  —  4:2:  =  1Q. 

4.  Passing  through  (5,  8),  and  ||  to  the  axis  of  x. 
(^   Passing  through  (5,  8),  and  !!  to  the  axis  of  y. 

6.  Passing  through  (3,  —13),  and  JL  to  the  line  y  —  4:X  —  7. 

7.  Passing  through  (2,  9),  and  J_  to  the  line  7y-f  23a:  — 5  =  0. 

8.  Passing  through  (0,  0),  and  _L  to  the  line  x-\-2y  —  l. 

9.  Perpendicular  to  the  line  bx  —  'Jy -\-l  =  0,  and  erected 
at  the  point  whose  abscissa  =  1. 


50 


ANALYTIC    GEOMETRY. 


10.  Perpendicular  to  the  line  y  — 3a:  =  2,  and  passing 
through  the  intersection  of  the  lines  x  —  y  =  l  and  2x -[• 
3y  =  7. 

Angles. 

52.  To  find  the  angle  formed  hy  the  lines  y  =  mx  +  h,  and 
y  =  m^x  -\-b\ 

Let  AB  and  CD  (Fig.  22),  represent  the  two  lines,  respec- 
tively, meeting  in  the  point  P. 

Let  the  angle  APC=<fi,  and  tan  <^  =  ?!.  By  Geometry, 
if>  =  y  —  y'.     Whence,  by  Trigonometry, 

m-m'  ^^^^ 


t 


1  +  mm' 

This  equation  determines  the  value  of  <f>. 


Fig.   22. 


Fig.    23. 


53.  To  find  the  equation  of  a  straight  line  passing  through 
a  given  point  (x^,  3/1),  and  making  a  given  angle  </>  luith  a  given 
line  y  =  mx-\-b. 

Let  the  required  equation  be 

y  -y,^m\x-x,) 

where  m'  is  not  yet  determined. 


THE   STRAIGHT    LINE.  61 

Since    the   required   line    may    lie    either   as  PQ   or    PR 
(Fig.  23),  ^ve  shall  have  ^by  §  b'S), 


m'  —  on  m  —  7n 


tan  <^  = or 


1  +  mm^         1  +  ^Mn' 

HI        7?i  -4-  tan  <6 
ence  7?i'  = ~~         ^  , 

1   ^=^?/t  tan  ^ 
and  the  required  equation  is 

rti  ±  tan  <|>    ,  .  m  -, 

v  —  yi  =  z-— — r-^(^  — a^i»  [11] 

1  :f  ^/i  tan  <j)  *-     -■ 

and  (as  Fig.  23  shows)  there  are  in  general  two  straight  lines 
satisfying  the  given  conditions. 


Ex.  14. 

1.  Find  the  angle  formed  by  the  lines  rr  +  2y  +  l  =  0  and 
ar-3y-4  =  p. 

The  two  slopes  are  —  h  and  |.  If  we  pvit  m  =  —  i,  vi'  =  ^,  we  obtain 
<•-— 1,  ^  =  135°.  If  weput  m  =  ^,  m'  =  — I,  weget  i  =  l,  0  =  45°.  Show 
that  both  these  results  are  correct. 

Find  the  tangent  of  the  angle  formed  by  the  lines 

2.  3a;-4y  =  7  and  2:r-y  =  3. 

3.  2a:  +  3y  4-4  =  0  and  3a;  +  4y  +  5  =  0. 

4.  y  — 01x^=1  and  2{y  —  l)=-nx. 

Find  the  angle  formed  by  the  lines 

5.  x-\-y  =  \  and  y  =  .'r-]-4. 

6.  y  +  3  =  2a:  and  y-f  3;r  =  2. 

7.  2:r  +  3y  +  7  =  0  and  5.T-2y  +  4  =  0. 

8.  6a:  =  2y-f-3  and  y-3.r  =  10. 

9.  a;  +  3  =  0  and  y  —  V3.r  +  4  =  0. 


62  ANALYTIC    GEOMETRY. 

10.  Discuss  equation  [11]  for  the  cases  where  fp  =  ^°  and 
<^  =  90°. 

Note,  The  learner  should  try  to  solve  the  next  five  exercises  directly, 
without  using  equation  [11]  ;  then  verify  the  result  by  means  of  [11]. 

Find  the  equation  of  a  straight  line 

11.  Passing  through  the  point  (3,  5),  and  making  the  angle 
45°  with  the  line  2a;-3y  +  5  =  0. 

12.  Passing  through  the  point  (—  2,  1),  and  making  the 
angle  45°  with  the  line  2?/  =  6  — 3a;. 

13.  Passing  through  that  point  of  the  line  y  =  2.r  — 1  for 
wdiich  X-—2,  and  making  the  angle  30°  with  the  same  line. 

14.  Passing  through  (1,  3),  and  making  the  angle  30°  with 
the  line  :r-2?/  +  l  =  0. 

15.  Prove  that  the  lines  represented  by  the  equations 

Axi-B2/-i-C=0,      A'x-i-B'y  +  C'  =  0 
are  parallel  if  AB'  =  A'B  ;  perpendicular,  if  A  A'  =  ~BB'. 

16.  Given  the  equation  3a;  +  4y+6  =  0;  show  that  the 
general  equations  representing  (i.)  all  parallels  and  (ii.)  all 
perpendiculars  to  the  given  line  are 

(i.)     3a;  +  4y  +  ^=0. 
(ii.)     ^x-2>y  +  K=0. 

17.  Deduce  the  following  equations  for  lines  passing 
through  (.Ti,  3/1)  and  (i.)  parallel,  (ii.)  perpendicular,  to  the 
line  y  =  mx-\-  h. 

(i.)     y  —  mx  =  3/1  —  r)iXx. 
(ii.)     my -\- X  =  m2j^-\- x^. 

18.  Write  the  equations  of  3  lines  parallel,  and  3  lines  per- 
pendicular, to  the  line  2a;  +  3?/-j-l  =  0. 


THE    STRAIeillT    LINE.  53 

10.    Among  the  following  lines  select  parallel  lines  ;  per- 
pendicular lines  ;  lines  neither  parallel  nor  perpendicular  : 
(i.)  2a;-f-3y-l  =  0.  (v.)  x-y  =  1. 

(ii.)  3a:-2y  =  20.  (vi.)  5(a;  +  2/) -11  =  0. 

(iii.)  4.^  +  6^  =  0.  (vii.)  a;  =  8. 

(iv.)  12:r  =  8y  +  7.  (viii.)  y  +  10  =  0. 

20.  Prove  that  the  angle  <^,  between  the  lines 

Ax-\-By-^G  =  ^  and  ^'o; +  ^'y  +  C"  =  0, 

is  determined  by  the  equation 

,      ^      AB'-A'B 
''"""^^'AA'  +  BB'' 

21.  From  the  preceding  equation  deduce  the  conditions  of 
parallel  lines  and  per^^endicular  lines  given  in  No.  15,  ]d.  52. 

Find  the  equation  of  a  straight  line 

22.  Parallel  to  2a;  +  3y  +  6  =  0,  and  passing  through  (5,  7). 

23.  Parallel  to   2a; +  y  — 1  =  0,  and  passing  through   the 
intersection  of  3a;  +  23/  — 59  =  0  and  5a:  — 7y  +  6  =  0. 

24.  Parallel   to   the   line  joining  (—2,  7)   and  (~4,  —  5), 
and  passing  through  (5,  3). 

25.  Parallel  to  y  =  '}nx-\-h,  and  at  a  distance  d  from  the 
origin. 

26.  Perpendicular  to  Ax  -f  By  -f  (7=  0,  and  cutting  an  in- 
tercept h  on  the  axis  of  y. 

X        1J 

27.  Perpendicular  to  -  +  t  =  1,  and  passing  through  (a,  h). 

X        1/ 

28.  Making   the    angle    45°   with  -+j  =  l,    and   passing 
through  {a,  0).  * 

29.  Show  that  the  triangle  whose  vertices  are  the  points 
(2,  1),  (3,  -2),  (-4,  -1)  is  a  right  triangle. 


54  ANALYTIC    GEOMETRY. 

30.  The  vertices  of  a  triangle  are  (—1,-1),  (—3,5), 
(7,  11).  Find  the  equations  of  its  altitudes.  Prove  that 
the  altitudes  meet  in  one  point. 

31.  Find  the  equation  of  the  perpendicular  erected  at  the 
middle  point  of  the  line  joining  (5,  2)  to  the  intersection  of 
x-\-2y-ll  =  0  and  9:i-- 2y  +  59  =  0. 

32.  Find  the  equations  of  the  perpendiculars  erected  at  the 
middle  points  of  the  sides  of  the  triangle  whose  vertices  are 
(5,  —7),  (1,  11),  (—4,  13).  Prove  that  these  perpendiculars 
meet  in  one  point. 

33.  The  equations  of  the  sides  of  a  triangle  are 
,^^4-7/+l  =  0,     3a:  +  5y  +  ll=0,     :i'-f2y  +  4  =  0. 

Find  (i.)  the  equations  of  the  perpendiculars  erected  at  the 
middle  points  of  the  sides  ;  (ii.)  the  co-ordinates  of  their  com- 
mon point  of  intersection  ;  (iii.)  the  distance  of  this  point  from 
the  vertices  of  the  triangle. 

34.  Show  that  the  straight  line  passing  through  {a,  I)  and 
((?,  d)  is  perpendicular  to  the  straight  line  passing  through 
(5,  —a)  and  (cZ,  —c). 

35.  What  is  the  equation  of  a  straight  line  passing  through 
(.Ti,  3/1),  and  making  an  angle  <^  with  the  line  Ax-\-By-\-C  =  0'^ 

Distances. 

54.  Find  the  distance  from  the  point  (—4,  1)  to  the  line 
3.?;-4?/  +  l  =  0.  Ans.   3. 

The  required  distance  is  the  length  of  the  perpendicular  let 
fall  from  the  given  point  to  the  given  line.  The  first  method 
that  occurs  for  solving  the  problem  is  to  form  the  equation_^f_ 
this  perpendicular,  find  its  intersection  witli,  the  given  linej^ 
andlhen  compute  the  distance  from  this  jntei^section^o  the 
given  point. 

Let  this  method  be  followed  in  solving  the  above  problem 
and  the  first  five  problems  of  Ex.  15. 


THE    STRAIGHT    LINE. 


65 


55.  To  find  the  distance  from  the  point  (xy,  y^)  to  the  line 
Ax^-Bij'-\-C=Q). 

Let  F  (Fig.  24)  represent  the  given  point  {x^,  y^),  and  AB 
the  given  line  Ax -\- By +  C  =-0,  Draw  PS  J_  to  AB,  and 
PM 1,  to  the  axis  of  :r  and  meeting  AB  in  a  point  P.  Let 
d  denote  the  required  distance  PS.     Then,  by  Trigonometry, 

d=PP  cos  PPS  =  PR  cos  SAX=  PR  cos  y. 


Fig.  24. 


Since  R  is  in  the  given  line,  and  021=  x^, 


RM  = 


Axy-C 
B 


Therefore     PR  -  MP  -  MR  =  ^•^t+^.Vi  +  ^. 

B 

To  find  the  value  of  cosy,  we  may  use  the  relations 


cos^y  +  sin-y  =  1,  and  - 

Eliminating  sin  y,  we  obtain 
B 


sm  y 
cosy 


cos  y  =  rh 


^A'  +  B' 
Substituting  these  values  of  PR  and  cosy,  we  have 


d 


Ax,  4-  gf/i  +  C^ 


[12] 


56  ANALYTIC    GEOMETRY. 

So  long  as  we  are  concerned  with  a  single  distance,  there  is 
no  occasion. for  the  use  of  both  signs,  and  we  should  choose 
that  sign  which  will  make  d  positive. 

Hence,  to  find  the  distance  from  the  point  {x^,  yj  to  the  line 
Ax-\-By^C  =  0,  we  have  as  a  practical  rule  :  Write  x^^  for 
X  and  ?/i  for  y,  and  divide  the  value  of  the  resulting  expression 
hy  V^M^. 

Solve  this  problem  when  the  given  point  is  assumed  to  be 
at  Q  (see  Fig.  24)  on  the  same  side  of  the  line  as  the  origin. 

What  is  the  value  of  d  when  the  given  j)oint  is  (i.)  the 
origin,  (ii.)  in  the  given  line? 

56.  To  find  the  distance  from  the  point  [x^,  y^  to  the  line 
X  cos  a-\-y  sin  a  —p  =  0. 

Let  the  equation  of  the  line  through  the  given  point,  par- 
allel to  the  given  line,  be  (§  41) 

X  cos  a -f- y  sina— p'  =  0. 

Since  (xi,  3/1)  is  in  this  line, 

r^i  cos  a  +  3/1  sin  a  —  ^9' =  0. 
Therefore  p'  =  Xi  cos  a  +  2/i  sin  a. 

'Now  p  and^:*'  are  the  distances  from  the  origin  to  the  given 
line,  and  its  parallel,  respectively  ;  therefore,  if  d  denote  the 
required  distance, 

d=  ±  {p'  —75)  =  =t  [xi  cos  a  +  ?/i  sin  a  ~ p)  ; 
and  in  general  we  should  choose  that  sign  which  will  make  d 
positive  :  the  j^ositive   sign,  if  (x-^,  y^)  and  the  origin  are  on 
ojoposite  sides  of  the  given  line;  the  negative  sign,  if  they  are 
on  the  same  side. 

If,  then,  the.  equation  of  a  straight  line  is  reduced  to  the 
normal  form  x  cos  a  -f  y  sin  a  — |;  =  0,  the  distance  from  any 
point  (:ri,  yf)  to  the  line  is  found  simply  by  substituting  on  the 
left-hand  side  of  the  ecjuation  x^  for  x  and  y^  for  y,  and  then 
computing  the  value  of  the  expression. 


XnE    STIIAIGIIT    LINE.  57 

Ex.  15. 

1.  Find  the  distance  from  (1,  13)  to  the  line  Srr^y  — 5. 

2.  Find  the  distance  from  (8,  4)  to  the  line  ?/  =  22:  — 16. 

3.  Find  the  distance  from  the  origin  to  the  line  3a;-|-4?/— 20. 

4.  Find  the  distance  from  (2,  3)  to  the  line  2:i-  +  y  —  4  =  0. 

5.  Find  the  distance  from  (3,  3)  to  the  line  y  =  4.x'  — 9. 

6.  Prove  that  the  distance  from  the  point  (x^,  y^)  to  the  line 

y  =  onx  4-  5  is  ^ 

d=  -u  .Vi  —  ^»-'^i  —  ^^ 

VT+T/? 

that  sign  being  chosen  which  will  make  d  positive.     Express 
this  result  in  the  form  of  a  rule  for  practice. 

7.  Find  the  distances  from  the  line  3.'r-|-4?/  +  15  =  0  to 
the  following  points  :  (3,  0).  (3,  - 1),  (3,  -  2),  (3,  -  3),  (3,  -  4), 
(3,-5),  (3,  -6),  (3,  -7),  (0,  0),  (-1,  0),  (-2,  0),  (-3,  0), 
(-4,0),  (-5,0),  (-6,0). 

8.  Find  the  distances  from  (1,  3)  to  the  following  lines  : 

3.r  +  4y  +  15  =  0.  3:r  +  4y-    5  =  0. 

3:r  +  4y-f  10  =  0.  3a;  +  4y-10  =  0. 

3.r  +  4y+    5  =  0.  3:r  +  4y-15  =  0. 

3.r  +  4y  =0.  3:c  +  4y-20  =  0. 

Find  the  following  distances  : 

9.    From  the  point  (2,  —5)  to  the  line  y  —  3:r  =  7. 

10.  From  the  point  (4,  5)  to  the  line  4y  +  5a;  =  20. 

11.  From  the  point  (2,  3)  to  the  line  x  -\- 1/  =  1. 

12.  From  the  point  (0,  1)  to  the  line  3.T-3y  =  l. 

13.  From  the  point  (—1,  3)  to  the  line  3a;  +  4y+2  =  0. 


68  ANALYTIC    GEOMETRY. 

14.  From  the  origin  to  the  line  3:r  +  2?/  —  6  =  0. 

15.  From  the  point  (2,  —7)  to  the  line  joining  (—4,  1)  and 
(3,  2). 

16.  From  the  line  y  =  lx  to  the  intersection  of  the  lines 
y  —  ^x  —  4:  and  y=bx-\-2. 

17.  From  the  origin  to  the  line  a(x  —  a)  -\-  h{x  —  &)  =  0. 

18.  From  the  points  (a,  h)  and  (—  a,  —  h)  to  fhe  line 

a      0 

19.  From  the  point  (a,  ^)  to  the  line  ax  -\-  hy  =  0. 

20.  From  the  point  (A,  k)  to  the  line  Ax -\- By +  0  =  1). 

Find  the  distance  between  the  two  parallels  : 

21.  3a;  +  4y  +  15  =  0  and  3a;  +  4z/  +  5  =  0. 

22.  3:^  +  4?/+15  =  0  and  3.^  +  4?/  — 5  =  0. 

23.  Ax-i-By +  C=0  &nd  Ax  +  By-}-C'  =  0. 

24.  9^  +  3y-7=0  and  9a;  +  3y-27=0. 

25.  y  =  5x  —  7  and  y  =  bx  -}-d. 

26.  ■^  +  f  =  2and^  +  f=J. 
a      0  a      0      A 

27.  Show  that  the  locus  of  a  point  whicli  is  equidistant 
from  the  lines  3a;  +  4y  — 12  =  0  and  4:r  +  o?/-24  =  0  con- 
sists of  two  straight  lines.  Find  their  equations,  and  draw  a 
figure  representing  the  four  lines. 

28.  Show  that  the  locus  of  a  point  which  so  moves  that 
the  sum  of  its  distances  from  two  given  straight  lines  is  con- 
stant is  a  straight  line. 


THE    STRAIGHT    LINE.  59 

« 

Areas. 

57.  Find  the  area  of  the  triangle  whose  vertices  are  the 
points  (2,  1),  (3,  -  2),  (-  4,  - 1).  Ans.  10. 

It  is  shown  in  Elementary  Geometry  that  the  area  of  a 
triangle  is  equal  to  one-half  the  product  of  its  base  and  its 
altitude ;  hence  this  problem  may  be  solved  by  performing 
the  following  operations  : 

(i.)  Find  the  length  of  one  side  chosen  as  base, 
(ii.)  Find  the  equation  of  the  altitude, 
(iii.)  Find  the  intersection  of  the  base  and  the  altitude, 
(iv.)  Find  the  length  of  the  altitude, 
(v.)  Multiply  this  length  by  one-half  the  base. 

Let  the  first  five  problems  of  Ex.  16  be  solved  in  this  way. 

58.  I'tnd  the  area  of  a  triangle,  having  given  its  vertices 
(^■i,  2/i),  {x,,  y,),  (.!;.,  y,). 

Solution  I.  If  we  take  as  base  the  line  joining  {x\  y,) 
and  {x-i,  y-i),  then 

base  --  V(y2  —  yO'  +  {x^  —  x{)\ 

The  altitude  is  the  distance  from  (.1-3,  yg)  to  the  base. 
Writing  in  equation  [12]  (p.  55)  x^  for  x^,  3/3  for  3/1,  and  for 
A,  B,  C  the  values 

^^y-i  — 3/1,  ^=-  —  0^2  —  ^1),  0  =  X2yi  —  x,7jo, 
obtained  by  equating  co-eflicients  in  [4]  and  [9]  (pp.  37,  43), 
we  get 

altitude  =  (.V^  ~  .vO^s  —  (^2  -  ^1)^3  +  ^2^1  —  ^1^2 
V(?/,-yxJ^-f(x,-^)^ 

The  area  of  the  triangle  =  i  base  X  altitude  ;  therefore 
area  =  J  [(y.,  —  yO^s  —  (^2  —  oc,)y^  +  x^y^  —  x^y^], 
which  may  be  written  more  symmetrically  thus : 

area  =  I  [x^{y.,  —  y.)  -f-  x.{y-,  —  y,)  +  x,{y,  —  y,)\.    [13] 


60 


ANALYTIC    GEOMETRY. 


Solution  II.  Let  PQR  (Fig.  25)  be  the  given  triangle, 
and  let  the  co-ordinates  of  PQR  be  x^y^,  x^y-i,  x^y^,  respectively. 
Drop  the  perpendiculars  PM,  QN,  EL  ;  then 

area  PQR  =  PQN3I+  RLNQ  -  PMLR. 
By  Geometry, 

PQNM=  ^M]V(PM+  QJST) 

Similarly, 

RLNQ  =  \{x^  -  X.)  (?/3  +  2/2), 
PMLR  =  ^{x,  -  X,)  {y,  +  2/0. 
Substituting  these  values,  we  have 

area  PQR  =  l[{x^  —  x^)  (?/,  -f  y^)  +  {x^  —  x^)  (3/3  +  y^) 

-  (x,  -  X,)  (2/3  +  yO] 
=  i  [•^2yi  —  ^12/2  +  ^'33/2  —  ^23/3  +  ^13/3  —  ^32/i] 
=  i[^i(y2  —  2/3)  +  ^2(^3  -  2/1)  +  ^3(yi  -  ^2)]- 


0 


Jl/ 


iv 


LX 


Fig.  25. 


Ex.  16. 
Find  the  area  of  the  triangle  whose  vertices  are  the  points 

1.  (0,  0),  (1,  2),  (2,  1). 

2.  (3,4),  (-3,-4),  (0,4). 

3.  (2,  3),  (4,-5),  (-3,-6). 

4.  (8,3),  (-2,3),  (4,-5). 

5.  (a,0),  (-a,0),  (0,1). 


THE   STRAIGHT    LINE.  61 

6.  Compare  the  formula  for  the  area  of  a  triangle  with  the 
result  obtained  by  solving  No.  68,  p.  42.  What,  then,  is  the 
geometric  meaning  of  that  result  ? 

Find  the  area  of  the  figure  having  for  vertices  the  points : 

7.  (3,  5),  (7,  11),  (9,  1). 

8.  (3, -2),  (5,  4),  (-7,  3). 

9.  (-1,2),  (4,4),  (6,-3). 

10.  (0,  0),  (x„  y,l  (t„  y,)- 

11.  (2,-5),  (2,8),  (-2,-5). 

12.  (10,5),  (-2,5),  (-5,-3),  (7,-3). 

13.  (0,  0),  (5,  0),  (9,  11),  (0,  3). 

14.  (a,  1),  (0,  b\  (c,  1). 

15.  (a,  h),  (b,  a),  (c,  c). 

16.  (a,  5),  (5,  a),  (c, -c). 

17.  Find   the   angles  and  the  area  of  the  triangle  whose 
vertices  are  (3,  0),  (0,  3V3),  (6,  3 V3). 

What  is  the  area  contained  by  the  lines 

18.  07  =  0,     ?/  =  0,     5a:  +  4y  =  20? 
l^.    x-{-y  =  l,     x  —  y  =  0,     y  =  0? 

20.  :r  +  2y=:5,     2x-\-y  =  7,     y  =  a:  +  l? 

21.  x  +  y  =  0,     x  =  9/,     y=3a? 

22.  y  =  Sx,     7/  =  7x,     7/'—c? 

2d.  x  =  0,     y  =  0,     :r-4  =  0,     y  +  6^0? 

24.  3a:  +  y  +  4-0,     3:r-5y  +  34  =  0,     3.r-2y  +  l-0? 

25.  ^-5y  +  13=:0,     5:r  +  7y  +  l  =  0,     3:f  +  y-9  =  0? 

26.  ^'  —  y  =  0,     x-^-y  =  0,     x~y  =  a,     x  -\-  y  =  b'l 


62  ANALYTIC    GEOMETRY. 

Find  the  area  contained  by  the  lines: 

27.  x  =  0,     y  =  0,     y  =  mx-\-b. 

28.  x  =  0,     y  =  0,     -  +  f  =  l. 

a      0 

29.  x  =  0,     7/  =  0,     Ax  +  By  +  C=0. 

30.  y  =  3p;-9,     y  =  2>x  +  b,     2y  =  a:-6,     23/  =  a;-fl4. 

31.  What  is  the  area  of  the  triangle  formed  by  drawing 
straight  lines  from  the  point  (2,  11)  to  the  points  in  the  line 
y  =^bx  —  6  for  which  ^^^i  =  4,  ajj  =  7  ? 

Ex.  17.     (Review.) 

1.  Deduce  equation  [7],  p.  39,  from  equation  [6]. 

2.  The  equation  y  =  mix  -f  ^  is  not  so  general  as  the  equa- 
tion Ax -\- By -\- C '=  ^ ,  because  it  cannot  represent  a  line 
parallel  to  the  axis  of  y.     Explain  more  fully. 

Determine  for  the  following  lines  the  values  of  a,  5,  y,  j),  and  a  : 

3.  :r=2.  6.    a;  +  V3y  =  2. 

4.  x^=y.  7.    a;  — Vo?/  =  2. 

5.  3/  +  l  =  V3(:r  +  2).  8.    ■\^d>x-y  =  2. 

9.  Find  the  equations  of  the  diagonals  of  the  figure  formed 
by  the  lines  3:r-y  +  9  =  0,  'ix  =  y-\,  5a;  +  3y=:18, 
5^'  +  3y  =  2.     What  kind  of  quadrilateral  is  it?     Why? 

10.  Find  the  distance  between  the  j^arallels  9a;  =  ?/  +  l 
and  ^x  =  y  —  7. 

11.  The  vertices  of  a  quadrilateral  are  (3,  12),  (7,  9). 
(2,  —  3),  (—  2,  0).     Find  the  equations  of  its  sides  and  its  area. 

12.  The  vertices  of  a  quadrilateral  are  (6,  —4),  (4,  4), 
(—4,  2),  (—8,  —6).  Prove  that  the  lines  joining  the  middle 
points  of  adjacent  sides  form  a  parallelogram.  Find  the  area 
of  this  parallelogram. 


THE    STRAIGHT    LINE.  63 

Find  the  equation  of  a  line  passing  through  (3,  4),  and  also 

13.  Perpendicular  to  the  axis  of  x. 

14.  Making  the  angle  45°  with  the  axis  of  :r. 

15.  Parallel  to  the  line  5.r  -f-  Gy  +  8  =  0. 

»1G.    Intercepting  on  the  axis  of  y  the  distance  — 10. 

H-'T.    Passing  through  the  point  half  way  between  (1,  —4; 
and  (-5,4). 

18.  Perpendicular  to  the  line  joining  (3,  4)  and  (—1,  0). 

Find  the  equations  of  the  following  lines  : 

19.  A  line  parallel  to  the  line  joining  (x^,  y,)  and  (x.^,  y^), 
and  passing  through  (x^,  y-^. 

20.  The  lines  passing  through  (S,  3),  (4,  3),  (-  5,  —  2). 

21.  A  line  passing  through  the  intersection  of  the  lines 
2a;  +  5y  +  8  =  0  and  3:f  — 4y  — 7=0,  and  ±  to  the  latter 
line. 

"     22.    A  lino  J_  to  the  line  4.r  —  y  =  0,  and  passing  through 
that  point  of  the  given  line  whose  abscissa  is  2. 

23.  A  line  li  to  the  line  3:^:  +  4y  =  0,  and  passing  through 
the  intersection  of  the  lines  :r  — 2y— a  =  0  and  a;-|-3y  — 2a  =  0. 

'^  24.    A  line  through  (4,  3),  such  that  the  given  point  bisects 
the  portion  contained  between  the  axes. 

25.  A  line  through  (.Tj,  yi),  such  that  the  given  point  bisects 
the  portion  contained  between  the  axes. 

26.  A  line  through  (4,  3),  and  forming  with  the  axes  in  the 
second  quadrant  a  triangle  whose  area  is  8. 

27.  A  line  through  (4,  3),  and  forming  with  the  axes  in  the 
fourth  quadrant  a  triangle  whose  area  is  8. 

28.  A  line  through  (—  4,  3),  such  that  the  portion  between 
the  axes  is  divided  by  the  given  point  in  the  ratio  5 :  3. 


04  ANALYTIC    GEOMETRY. 

29.  A  line  dividing  the  distance  between  (—3,  7)  and 
(5,-4)  in  the  ratio  4:7,  and  J_  to  the  line  joining  these 
points. 

30.  The  two  lines  through  (3,  5)  making  the  angle  45°  with 
the  line  2:r-3y-7-=0. 

31.  The  two  lines  through  (7,  —5)  which  make  the  angle 
45°  with  the  line  6:r  —  2y  +  3  =  0. 

32.  The  line  making  the  angle  45°  with  the  line  joining 
(7,  — 1)  and  (—3,  5),  and  intercepting  the  distance  5  on  the 
axis  of  X, 

•33.  The  two  lines  which  pass  through  the  origin  and  tri- 
sect the  portion  of  the  line  x-\-y=\  included  between  the 
axes. 

34.  The  two  lines  il  to  the  line  4.r  +  5?/  + 11=  0,  at  the 
distance  3  from  it. 

35.  The  bisectors  of  the  angles  contained  between  the  lines 
y  =  2a;  +  4,  y  =  %x-^^. 

Hi^sTT.  Every  point  in  the  bisector  of  an  angle  is  equidistant  from 
the  sides  of  the  angle. 


6.    The  bisectors  of  the  angles  contained  between  the  lines 


•o 


2a;-5?/  =  0,  4:r  +  3?/  =  l 


37.  The  two  lines  which  pass  through  (3,  12),  and  whose 
distance  from  (7,  2)  is  equal  to  V58. 

38.  The  two  lines  which  pass  through  (—2,  5),   and  are 
each  equidistant  from  (3,  —7)  and  (—4,  1). 

Find  the  angle  contained  between  the  lines: 

39.  y  +  3  =  2a;  and  y+3a:  =  2. 

40.  ?/  =  5a:  — 7  and  5?/  + a;- 3  =  0. 


THE    STRAIGUT    LINE.  65 

Find  the  distance  : 

41.  From  the  intersection  of  the  lines  3^'  +  2y  +  4  =  0, 
2^-  +  5y  +  8=-0  to  the  line  y  =  5a:  +  6. 

42.  From  the  point  (h,  Jc)  to  the  line  -  +  ^^  =  1. 

43.  From  the  origin  to  the  line  hx  +  ^y  =  <?^ 

44.  From  the  point  (a,  0)  to  the  line  y  =  vix  +  —  • 

Find  the  area  included  between  the  following  lines : 

45.  X  ■=  y,     x-\-  y  ■=(),     x  =  c. 

46.  x-i-y  =  k,     2x  =  y  +  k,     2y  =  x  +  k 

47.  ^  +  1  =  1.     y  =  2.r  +  5,     x  =  2y  +  a. 

48.  y  =  4.r  +  7  and  the  lines  which  join  the  origin  to  those 
points  of  the  given  line  whose  ordinates  are  —  1  and  19. 

49.  The  lines  joining  the  middle  points  of  the  sides  of  the 
triangle  formed  by  the  lines  x  —  5y-l-ll=--0,  11. t +  6?/  — 1  =  0, 
x  +  y  +  4.^0. 

50.  Find  the  area  of  the  quadrilateral  whose  vertices  are 
(0,  0),  (0,  5),  (11,  9).  (7,  0). 

51.  AVhat  point  in  the  line  5^7  — 4y  —  28  =  0  is  equidis- 
tant from  the  points  (1,  5)  and  (7,  —  3)  ? 

52.  Prove  that  the  diagonals  of  a  square  are  J.  to  each 
other. 

53.  Prove  that  the  line  joining  the  middle  point  of  two  sides 
of  a  triangle  is  parallel  to  the  third  side. 

54.  What  is  the  geometric  meaning  of  the  equation 
xy  =  0? 


66  ANALYTIC    GEOMETRY. 

55.  Show  that  the  three  points  (3a,  0),  (0,  2>h),  {a,  2b)' 
are  in  a  straight  line. 

56.  Show  that  the  three  lines  5x -{-Si/ —  7  =0,  3a:  — 42/ 
— 10  =  0,  and  ^  -f  2y  =  0  meet  in  a  point. 

57.  What  must  be  the  value  of  a  in  order  that  the  three 
lines  3:r  +  3/— 2  =  0,  2a;  — y  — 3  =  0,  and  aa:  +  2y  —  3  =  0 

may  meet  in  a  point  ? 

What  straight  lines  are  represented  by  the  equations : 

58.  x''  +  (a-h)x-ab  =  0? 

59.  xij  -^-hx-^-  ay  +  «^  =  0  ? 

60.  x\j=^xif1 

61.  14a;^-5.Ty  — y'  =  0? 

In  the  following  exercises  prove  that  the  locus  of  the  point 
is  a  straight  line,  and  obtain  its  equation. 

62.  The  locus  of  the  vertex  of  a  triangle  having  the  base 
and  the  area  constant. 

63.  The  locus  of  a  j)oint  equidistant  from  the  points  {x^,  ?/i). 
and  (.T.,  y.?}. 

64.  The  locus  of  a  point  at  the  distance  d  from  the  line 
Ax^By^C  =  ^. 

65.  The  locus  of  a  point  so  moving  that  the  sum  of  its 
distances  from  the  axes  shall  be  constant  and  equal  to  Tc. 

66.  The  locus  of  a  jioint  so  moving  that  the  sum  of  its 
distances  from  the  lines  Ax^By-\-C=^,  ^'a- +^'y  +  ^' =  0 
shall  be  constant  and  equal  to  Iz. 

67.  The  locus  of  the  vertex  of  a  triangle,  having  given  the 
base  and  the  difference  of  the  squares  of  the  other  sides. 


THE   STRAIGUT    LINE.  67 


SUPPLEMENTARY   PROPOSITIONS. 

Lines  passing  through  One  Point. 

59.  If  S  =  0,  S'  —0  represent  the  equations  of  any  two  loci 
with  the  terms  all  transposed  to  the  left-hand  side,  and  k  de- 
notes an  arbitrary  constant,  then  the  locus  represented  hy  the 
equation  /S -^  kS' =  0  2^(^sses  throurjh  every  point  common  to 
the  tivo  given  loci. 

For  it  is  plain  that  any  co-ordinates  which  satisfy  the  equa- 
tion S=0,  and  also  satisfy  the  equation  jS'  =  0,  must  like- 
wise satisfy  the  equation  S-\-  kS'  =  0. 

For  what  values  of  k  will  the  equation  ^  +  kS'  =  0  repre- 
sent the  lines  jS==0  and  /S"  =  0,  respectively? 

60.  Mnd  the  equation  of  the  line  joining  the  point  (3,  4)  to 
the  intersection  of  the  lines 

3:i;-2y-fl7=0  and  a.-  +  4y-27=0. 

The  method  of  solving  this  question  which  first  occurs  is 
to  find  the  intersection  of  the  given  lines  and  then  apply 
equation  [4],  p.  37. 

Another  method,  almost  equally  obvious,  is  to  employ  equa- 
tion [5],  which  gives  at  once 

y-^  =  m{x-Z), 
and  then  determine  m  by  substituting  for  x  and  y  the  co-ordi- 
nates of  the  intersection  of  the  given  lines. 

The  following  method,  founded  on  the  princi|)le  stated  in 
§  59,  is,  however,  sometimes  preferable,  on  account  of  its 
generality  and  because  it  saves  the  labor  of  solving  the  given 
equations.  According  to  this  principle,  the  required  equation 
may  be  immediately  written  in  the  form 

3a;  -  2y  +  17-f  /<^  +  4y  -  27)  =  0. 


68  ANALYTIC    GEOMETRY. 

And  since  tlie  line  passes  through  (3,  4),  we  must  have 
9  -  8  + 1 7  +  ^<3  + 16  -  27)  =  0, 

whence  ^  =  -• 

4 

Therefore    12a:-8y+68  +  9^  +  36y-243  =  0, 
or  3:r  +  4y  — 25--0. 

This  is  the  equation  of  the  required  line. 

61.  If  the  equations  of  three  straight  lines  are 

and  ive  can  find  three  constants,  I,  on,  n,  so  that  the  relation 
l{Ax  +i??/ +  C)  +  m(A'x+B'y+  C)  +  n(A"x+II''7/-\-  C")  =  0 
is  identically  true,  that  is,  true  for  all  values  of  x  and  y,  then 
the  three  lines  tneet  in  a  point. 

For  if  the  co-ordinates  of  any  point  satisfy  any  two  of  the 
equations,  then  the  above  relation  shows  that  they  will  also 
satisfy  the  third  equation. 

62.  To  find  the  equation  of  a  straight  line  passing  through 
the  intersection  of  the  two  lines 

Ax-hI!2j  +  C=0,     A'x  +  J5'iji-C'  =  0, 
and  bisecting  the  angle  between  them. 

There  are  evidently  two  bisectors  :  one  bisecting  the  angle 
in  which  the  origin  lies  ;  the  other  bisecting  the  supplementary 
angle. 

The  simplest  way  to  obtain  their  equations  is  to  express 
algebraically  the  fact  (proved  in  Geometry)  that  any  point 
(.r,  y)  of  the  bisector  is  equidistant  from  the  sides  of  the  angle. 
Hence  from  equation  [12],  p.  55,  we  immediately  obtain  the 
equation       ^^  _|_ ^^  4.  ^  ^  _^  Ax  -¥B'y  +  C ^ 

which  represents  both  bisectors  if  we  use  both  signs  on  the 
ricrht-hand  side. 


THE    STRAIGHT    LINE.  G9 

In  order  to  distinguish  between  the  bisectors,  it  is  necessary 
to  pay  attention  to  the  sign  of  the  distance  from  a  straight 
line  to  a  point.  We  see  from  equation  [12]  that  this  dis- 
tance changes  sign  when  the  point  crosses  the  line  ;  let  it  be 
agreed  that  distances  measured  to  points  on  the  origin  side  of 
the  line  shall  be  considered  positive,  and  that  distances  meas- 
ured to  points  on  the  side  remote  from  the  origin  shall  be 
considered  negative. 

"Now  the  distance  from  the  line  Ax -^r  By -\- C  =^  ^  to  the 
origin  itself  is  p 


and  in  order  that  this  may  be  always  positive,  we  must  place 
before  it  the  same  sign  as  that  of  C.  It  follows  that  equation 
[14]  will  represent  the  bisector  of  the  angle  in  which  the  origin 
lies  if  we  choose  that  sign  which  will  make  the  two  constant 
terms  alike  in  sign. 

If  we  choose  the  other  sign,  the  equation  of  course  will 
represent  the  bisector  of  the  supplementary  angle. 

63.  To  find  the  ^ equation  of  a  straight  line  passing  through 
the  intersection  of  the  two  lines 

X  cos  a  -f  y  sin  a  — ^  =  0,     X  cos  o!  -\-y  sin  a'  — p^  —  0, 
and  bisecting  the  angle  between  them. 

Taking  the  angle  which  includes  the  origin,  and  denoting 
by  {x,  y)  any  point  in  the  bisector,  we  have  immediately  for 
its  equation 

{x  cos  a  +  y  sin  a  — 7?)  —  (x  cos  a.'  -\-  y  sin  a  —  p^^  =  0. 

The  equation  of  the  bisector  of  the  supplementary  angle  is 

{x  cos  a -f- 2/  sin  a  — 'p)  +  {x  cos  o!  -\-y  sin  o!  —  pj)  =  0. 

It  may  be  shown  from  the  form  of  these  equations  that  the 
two  bisectors  are  perpendicular- to  each  other. 


70  ANALYTIC    GEOMETRY. 

Ex.  18. 

Find  the  equation  of  a  line  passing  l^hrough  the  intersection 
of  the  lines  3^^  +  2?/  +  17  =  0,  :r  +  4y-27--0,  and 

1.  Passing  also  through  the  origin. 

2.  Parallel  to  the  line  .r  +  2y  +  3  =  0. 

3.  Perpendicular  to  the  line  6x  —  5i/  --=  0. 

4.  Equally  inclined  to  the  two  axes. 

5.  Find  the  equation  of  a  line  parallel  to  the  line  x  =  y, 
and  passing  through  the  intersection  of  the  lines  y  =  2a:-l-l 
and  ?/  +  3:?:  =  ll. 

6.  Find  the  equation  of  a  straight  line  joining  (2,  3)  to  the 
intersection  of  the  lines 

2.'i;  +  3y  +  l=0  and  3a;-4y  =  5. 

7.  Find  the  equation  of  a  straight  line  joining  (0,  0)  to  the 
intersection  of  the  lines 

5a:  — 2y  +  3  =  0  and  lSx-}-i/  =  l. 

8.  Find  the  equation  of  a  straight  line  joining  (P,  11)  to  the 
intersection  of  the  lines 

2x-i-57/-S  =  0  and  Sx—4:i/=8. 

Find  the  equation  of  the  straight  line  passing  through  the 
intersection  of  the  lines  Ax -\- Bi/ -\- C  =  0  and  A'x -{- J5'i/ 
+  C*'  =  0,  and  also 

9.    Passing  through  the  origin. 

10.  Drawn  parallel  to  the  axis  of  x. 

11.  Passing  through  the  point  (x^,  y^. 

12.  Find  the  equation  of  a  straight  line  passing  through 
the  intersection  of  5a:  — 4y  +  3  =  0  and  7a: -f- lly  —  1  =  0, 
and  cutting  on  the  axis  of  y  an  intercept  equal  to  6. 


THE    STRAIGHT    LINE.  71 

13.  Find  the  equation  of  a  straight  line  passing  through 
the  intersection  of  ?/  =  7a:— 4  and  y— -  —  2:i-  +  5,  and  forming 
with  the  axis  of  x  the  angle  G0°. 

14.  The  distance  of  a  straight  line  from  the  origin  is  5  ; 
and  it  passes  through  the  intersection  of  the  lines  3.r— 2?/ 
+  11  =  0  and  6  a;  +  7  ?/  —  55  =  0.     What  is  its  equation  ? 

15.  What  is  the  equation  of  the  straight  line  passing 
through  the  intersection  of  bx  -j-  ay  ■=  ah  and  y  =  inx^  and 
perpendicular  to  the  former  line  ? 

Prove  that  the  following  lines  are,  concurrent  (or  pass 
through  one  point)  : 

16.  y  =  1x-\-\,     y  =  :r  +  3,     ?/  =  — 5.r  +  15. 

17.  4a:-2y-3=-0,     3:r-y  +  i  =  0,     5a;  — 2y-l  =  0. 

18.  2a:  — y==5,     3.r-y=-6,     4a:  — ?/  =  7. 

19.  What  is  the  value  of  ??^  if  the  lines 

-+7=1,     T  +  -  =  l,     y  =  mx 
ah  ha  ^ 

meet  in  one  point  ? 

20.  When  do  the  straight  lines  y  =  7nx  -\-h,  y  -=  m'o:  +  h\ 
y  —  m^^x  +  Z/"  pass  through  one  point  ? 

21.  Prove  that  the  three  altitudes  of  a  triangle  meet  in  one 
point. 

22.  Prove  that  the  perpendiculars  erected  at  the  middle 
points  of  the  sides  of  a  triangle  meet  in  one  point. 

23.  Prove  that  the  three  medians  of  a  triangle  meet  in  one 
point.  Show  also  that  this  point  is  one  of  three  points  of 
trisection  for  each  median. 

24.  Prove  that  the  bisectors  of  the  three  angles  of  a  triangle 
meet  in  one  point. 


72  ANALYTIC    GEOMETRY. 

25.  The  vertices  of  a  triangle  are  (2, 1),  (3,-2),  (-  4,  - 1). 
Find  the  lengths  of  its  altitudes.  Is  the  origin  within  or 
without  the  triangle  ? 

26.  The  equations  of  the  sides  of  a  triangle  are 

3a;  +  3/  +  4  =  0,     3.T-5y  +  34  =  0,     3.r  — 2y  +  l=0. 
Find  the  lengths  of  its  altitudes. 

What  are  the  equations  of  the  lines  bisecting  the  angles 
between  the  lines 

27.  3:r-4y  +  7  =  0  and  4.T-3y  +  17=0? 

28.  3:r-43/-9  =  0  and  12:r  +  5y-3  =  0? 

29.  y  =  2:r-4  and  2?/  =  x-\-10? 

30.  x-i-2/  =  2  Siud  x-i/  =  0? 

31.  y  =  7nx  4-  i>  and  y  =  vi'x  -}-h'  ? 

32.  Prove  that  the  bisectors  of  the  two  supplementary  angles 
formed  by  two  intersecting  lines  are  perpendicular  to  each 
other. 

Equations  eepeesenting  Straight  Lines. 

64.  A  homogeneous  equation  of  the  nth  degree  represents  n 
straight  lines  through  the  origin. 

Let  the  equation  be 

Ax^  +  Bx^'-'y  +  ar"-y  + +  Kif  -=  0. 

Dividing  by  ^y",  we  have 

M"+|AY-+^f-^r+ +  f=o. 

\yj       A\yJ          A\gJ  A 

If  n,  I't^  ^3 ^n  denote  the  roots  of  this  equation,  then  the 

equation,  resolved  into  its  factors,  becomes 


(i-'.)(i-)e-) e-) 


0. 


THE    STRAIGHT    LINE.  id 

and  therefore  is  satisfied  when  each  one  of  these  factors  is 
zero,  and  in  no  other  cases. 

Therefore  the  locus  of  the  equation  consists  of  the  n  straight 
lines  A  ri  A 

65.    To  find  the  angle  between  the  two  straight  lines  repre- 
sented by  the  equation  Ax^  +  Hxt/  -^-Bif  =  0. 

Solving  the  equation  as  a  quadratic  in  x,  we  obtain 

Hence  the  slopes  of  the  two  lines  are 

1A  ,  2A 


ni 


H-^H'-^AB  -H+^H'-^AB 

Therefore 

,      -JH'~^AB  ,      A 

7n7n 


B  B' 

and  (equation  [10],  p.  50) 

^  _  m-  m'  _  V//^  —  ^AB 
\-\-mm^  A+B 

66.  To  find  the  condition  that  the  general  equation  of  the 
second  degree  may  represent  two  straight  lines. 

AVe  may  write  the  most  general  form  of  the  equation  of  the 
second  degree  as  follows  : 

Ax'  ■\-Hxy-\-By''-\-Dx-^Ey  +  C=  0.  (1) 

In  order  that  this  equation  may  represent  two  straight  lines, 
it  must  be  equivalent  to  the  product  of  two  linear  factors  ; 
that  is,  equivalent  to  an  equation  of  the  form 

{Ix  +  my  +  n)  {px  -^qy-\-r)  =  0.  (2) 

Equating  coefficients  in  (1)  and  (2),  we  obtain 

lp  =  A,  mq=B,  nr~0, 

Iq  +  mp  =11,       lr-{-np=  I),       mr  +  nq  =  E. 


74  ANALYTIC    GEOMETRY. 

The  product  of  H,  D,  and  E  is 

IIDE  =  2  ImniDqr  +  lp{n^q^  +  ??i^'r^)  +  mq(J}r'^  +  ^^i?^) 

=  2^^C    +^(^^-2^(7)+i?(i)^-2^C) 
+  C(//'^-2^^). 
Hence  the  required  condition  is 

^ABC-  AE'  -  BE'  -  CH'  +  ^BE  =  0. 

Ex.  19. 

1.  Describe  the  position  of  the  two  straight  lines  repre- 
sented by  the  equation  Ax'""  +  ILiy  -\-  Bif  -\-Dj:  -\- Ey  ^  C^  0, 
where  (i.)  A=  11^ D  =  0,  (ii.)  B  =  JI--^ E=  0. 

2.  When  will  the  equation  aiy  -j-  bx  ^  cij  -\-  d  —-  0  repre- 
sent two  straight  lines  ? 

3.  Find  the  conditions  that  the  straight  lines  represented 
by  the  equation  Ax"^ -\- E'xi/ -\- B?j^  ^=  0  may  be  real;  imagi- 
nary ;  coincident ;  perpendicular  to  each  other. 

4.  Show  that  the  two  straight  lines  x'^  —  2xi/  sec  0  -\-  i/'  =  0 
make  the  angle  0  with  each  other. 

Show  that  the  following  equations  represent  straight  lines, 
and  find  their  separate  equations  : 

5.  x''~2x7/-d7/  +  2x-27/  +  l=0. 

6.  x'-4:xi/  +  57/-~6i/-{-9  =  0. 

7.  x'-4:Xi/-{-Si/  +  (di/-~9  =  0. 

8.  Show  that  the  equation  x"^ -j- x?/ ~  6 y"^ -^7 x -^  31 7/ — 18  —  0 
represents  two  straight  lines,  and  find  the  angle  between  them. 

Determine  the  values  of  E  for  which  the  following  equa- 
tions will  represent  in  each  case  a  pair  of  straight  lines.  Are 
the  lines  real  or  imaginary? 

9.  12x'-l0xi/~{-2f-}-Ux-5i/-{-K=0. 


THE    STRAIGHT    LINE.  75 

10.  12.rH-A:n/  +  2y^  +  lla:  — 5y  +  2=a 

11.  l:2.r^  +  36.ry  +  /r/  +  6:r-h6y  +  3  =  0. 

12.  For  what  value  of  K  does  the  equation  I{'xi/-\-bx 
+  3 y  +  2  =  0  represent  two  straight  lines  ? 

Problems  on  Loci  involving  Three  Variables. 

67.  A  trapezoid  is  formed  by  drawing  a  line  iKirallel  to  the 
base  of  a  given  triangle.  Find  the  locus  of  the  intersection  of 
its  diagonals. 

If  ABC  be  the  given  triangle,  and  we  choose  for  axes  the 
base  AB  and  the  altitude  CO,  the  vertices  A,  B,  C  may  be 
represented  in  general  by  {a,  0),  {b,  0),  (0,  <?),  respectively. 
The  equations  of  ^Cand  ^Care 

-  +  -  =  1    and    y  +  -  =  l. 
a      c  be 

Let  y  =  m  be  the  equation  of  the  line  parallel  to  the  base, 
and  let  it  cut  AC  in  D,  BC  in  E ;  then  the  co-ordinates  of 
D  and  E,  respectively,  are 

fac  —  am      \         -,    fbc  —  bin 
I ,  m      and      ,  m 


(1) 


\       c  J  \      ^ 

Hence  the  equation  of  the  diagonal  BB  is 
y     cm 

X  —  b      ac  —  am  —  be 
and  the  equation  of  the  diagonal  AE  is 

y     ^  cm  .^ 

x  —  a  be  —  bm  —  ac 
If  P  be  the  intersection  of  the  diagonals,  then  the  co-ordi- 
nates X  and  y  of  the  point  P  must  satisfy  both  (1)  and  (2)  ; 
by  solving  these  equations,  therefore,  we  obtain  for  any  pa,r- 
ticular  value  of  m  the  co-ordinates  of  the  point  P.  But  what 
we  want  is  the   algebraic  relation  which  is  satisfied  by  the 


76  ANALYTIC    GEOMETRY. 

co-ordinates  of  P,  ivJuitcver  the  value  of  m  may  he.  To  find 
this,  we  have  only  to  eliminate  m  from  equations  (1)  and  (2). 
By  doing  this  we  obtain 

2cx+{a  +  b)y  =  {a-^h)c, 

We  see  from  the  form  of  this  equation  that  the  required 
locus  is  the  line  which  joins  C  to  the  middle  point  'of  AB. 

Remakk.  The  above  solution  should  be  studied  till  it  is  understood. 
In  problems  on  loci  it  is  often  necessary  to  obtain  relations  which  in- 
volve not  only  the  x  and  2/  of  a  point  of  the  locus  which  we  are  seeking, 
but  also  some  third  variable  (as  m  in  the  above  example). 

In  such  cases  we  must  obtain  two  equations  which  involve  x  and  y 
and  this  third  variable,  and  then  eliminate  the  third  variable ;  the 
resulting  equation  wall  be  the  equation  of  the  locus  required. 

Ex.  20. 

1.  Through  a  fixed  point  0  any  straight  line  is  drawn,  meet- 
ing two  given  parallel  straight  lines  in  P  and  Q ;  through  P 
and  Q  straight  lines  are  drawn  in  fixed  directions,  meeting 
in  P.  Prove  that  the  locus  of  P  is  a  straight  line,  and  find 
its  equation. 

2.  The  hypotenuse  of  a  right  triangle  slides  between  the 
axes  of  X  and  y,  its  ends  always  touching  the  axes.  Find  the 
locus  of  the  vertex  of  the  right  angle. 

3.  Given  two  fixed  points,  A  and  P,  one  on  each  of  the 
axes  ;  if  U  and  V  are  two  variable  points,  one  on  each  axis, 
so  taken  that  0U+  OV=OA  +  OB,  find  the  locus  of  the 
intersection  of  A  V  and  P  U. 

4.  Find  the  locus  of  the  middle  points  of  the  rectangles 
which  may  be  inscribed  in  a  given  triangle. 

5.  If  PP\  QQ'  nre  any  two  parallels  to  the  sides  of  a  given 
rectangle,  find  the  locus  of  the  intersection  of  FQ  and  P'Q'. 


CHAPTER   III. 


THE    CIRCLE. 
Equations  of  the  Circle. 

68.  The  Circle  is  the  locus  of  a  point  which  moves  so  that 
its  distance  from  a  fixed  point  is  constant.  The  fixed  point 
is  the  centre,  and  the  constant  distance  the  radius,  of  the  circle. 

Note.  The  word  "  circle,"  as  here  defined,  means  the  same  thing  as 
"circumference  "  in  Elementary  Geometry.  This  is  the  usual  meaning 
of  "circle"  in  the  higher  branches  of  Mathematics. 

69.  To  find  the  equation  of  a  circle,  Jiaving  given  its  centre 
{ci,  b)  and  its  radius  r. 


Fig.   26. 

Let  C  (Fig.  2G)  be  the  centre,  and  P  any  point  (.r,  y)  of 
the  circumference.  Then  it  is  only  necessary  to  express  by 
an  equation  the  fact  that  the  distance  from  P  to  (7  is  constant, 
and  equal  to  r :  the  required  equation  evidently  is  (§  G) 

{jc  —  ay  +  (2/  —  by  =  r\  [15] 


78  ANALYTIC    GEOMETRY. 

If  we  draw  Ci?  li  to  OX,  to  meet  the  ordinate  of  P,  then 
we  see  from  the  figure  that  the  legs  of  the  rt.  A  CPPi,  are 
CR  =  x-a,  PB=--y-h. 

If  the  origin  be  taken  at  the  centre,  then  a  =  5  =  0,  and  the 
equation  of  the  circle  is 

This  is  the  simplest  form  of  the  equation  of  a  circle,  and 
the  one  most  commonly  used. 

If  the  origin  be  taken  on  the  circumference  at  the  point  A, 
and  the  diameter  AB  be  taken  as  the  axis  of  x,  then  the  centre 
will  be  the  point  (r,  0).  Writing  r  in  place  of  a,  and  0  in  place 
of  h  in  [15],  and  reducing,  we  obtain 

X- H- 2/2  =  2  rx.  [17] 

Why  is  this  equation  without  any  constant  term  ? 

70.  To  find  the  condition  that  the  general  equation  of  the  second 
degree,  Ax- +  Hxy  ~^Bif  ^  Dx  +  Ey -{-0=0,  (1) 

shall  represent  a  circle. 

If  possible,  let  it  be  the  circle  whose  centre  is  the  point 
(a,  h)  and  whose  radius  is  r.  The  equation  of  this  circle  has 
been  found  to  be 

(^x-ay^{y-hy  =  r\ 
or  x'^y''-2ax-2hy  +  a'  +  h''-r''  =  0.  (2) 

Equating  corresponding  coefficients  and  constant  terms,  we 
have  A^i^  B  =  \  11=0, 

I)  =  -2a,     B=-2b,     C  =  a'  +  h'-r\ 
Since,   if  ^  =  P  in   equation  (1),  both   A   and  B  can  be 
reduced  to  unity  by  division,  the  two  conditions  necessary  in 
order  that  equation  (1)  may  represent  a  circle  are 

A  =  B   and    ^=0; 
and  the  general  equation  of  a  circle  may  be  written 
x^  -\-  y'  +  Dx  +  Ey  +  C=0. 


THE    CIRCLE.  79 

The   co-ordinates  of  the   centre   and   the   radius  have   the 


values 


D       y  E  , 


Ex.  21. 
Find  the  equation  of  the  circle,  taking  as  origin 

1.  The  point  B  (Fig.  26). 

2.  The  point  D  (Fig.  26). 

3.  The  point  E  (Fig.  26). 

Write  the  equations  of  the  following  circles  : 

4.  Centre  (5,  —  3),  radius  10. 

5.  Centre  (0,  -2),  radius  11. 

6.  Centre  (5,  0),  radius  5.  ■ 

7.  Centre  (—  5,  0),  radius  5. 

8.  Centre  (2,  3),  diameter  10. 


9.    Centre  (/i,  h),  radius  VA'  -f  h^. 

10.  Determine  the  centre  and  radius  of  the  circle 

.t'  +  y"  -10:r  +  12y  +  25  =  0. 

In    this   case  D^-IO,  ^=12,   C=  25  (see  §  70).      Therefore  a  =  5, 
5  = -6,  r  =  V25  +  3G-25  =  6. 

Determine  the  centres  and  radii  of  the  followinsf  circles  : 

o 

11.  a;^  +  2/--2:r-4y  =  0.  17.  6:r- —  2y(7  -  3^/)  =  0. 

12.  3a;2+3/-5a:-72/+l=0.  18.  x'-^if=^h\ 

13.  x'-\-y'-'^x  =  ^.  19.  {xAryy-^{x-yy='^h\ 

14.  :r2  +  y2  +  8:c  =  0.  20.  x^ -\- if -=  a^ -\- h\ 

15.  a;=  +  /-8y  =  0.  21.  x^  ^y""  ^Icix^lc). 

16.  a;'  +  y2a-83/  =  0.  22.  x"  ^  y""  =  hx  ■\- hj , 


80  ANALYTIC    GEOMETRY. 

23.  When  are  the  circles  x"  +  y"" +Dx-^ Ey -\-C=0  and 
x"  +  y'^-^Ux  +^'?/  +  <^'  =  0  concentric  ? 

24.  What  is  the  geometric  meaning  of  the  equation  {x  —  df 
-i-(y-by  =  0? 

25.  Find  the  intercepts  of  the  circles 

(i.)  x''}-y'-8x~8y  +  1=0, 

(ii.)  x'-\-y'-Sx  —  Sy  +  l6  =  0, 

(in.)  x'''^y'-8x~8y  +  20  =  0. 

Putting  3/ =  0  in  each  case,  we  have  in  case  (i.)  a;^  — 8 a; +  7=0, 
whence  x  =  l  and  7;  in  case  (ii.)  a-'^  — 8a;  + 16  =  0,  whence  a;  =  4;  in 
case  (iii.)  x^  —  8  a;  +  20  =  0,  whence  a;  =  ±  V—  4. 

Putting  a;  =  0  in  each  case,  we  obtain  for  y  values  indentical  with  the 
above  values  of  x. 

The  geometric  meaning  of  these  results  is  as  follows  : 

Circle  (i.)  cuts  the  axis  of  a;  in  the  points  (1,  0),  (7,  0),  and  the  axis  of 
y  in  the  points  (0, 1),  (0,  7). 

Circle  (ii.)  touches  the  axis  of  x  at  (4,  0),  and  the  axis  of  y  at  (0,  4). 

Circle  (lii.)  does  riot  meet  the  axes  at  all. 

This  is  the  meaning  of  the  imaginary  values  of  x  and  y  in  case  (iii.). 

If,  however,  we  wish  to  make  the  language  of  Geometry  conform 
exactly  to  that  of  Algebra,  then  in  this  case  we  should  not  say  that 
the  circle  does  not  meet  the  axes  at  all,  but  that  it  meets  them  in  imagi- 
nary points  ;  just  as  we  do  not  say  that  the  equation  a;^  —  8  a;  +  20  =  0 
has  no  roots,  but  that  it  has  two  imaginary  roots. 

Find  the  centres,  radii,  and  intercepts  on  the  axes  of  the 


following  circles 

'• 

26. 

x'  +  f- 

5x- 

-72/  +  6  =  0. 

27. 

x'  +  y'- 

-12x 

-4?/  +  15  = 

0. 

28.  x'  +  7/'-4:X-  8y  =  0. 

29.  x'-{-y'-6x  +  4:y-\-4:  =  0. 

30.  x^-\-y'-]-22x  —  18y  +  b7=0. 


THE    CIRCLE.  81 

31.  Under  what  conditions  will  the  circle  x' -\' if -\- Dx 
-}-^y-]-(7=0  (i.)  touch  the  axis  of  x1  (ii.)  touch  the  axis 
of  y  ?  (iii.)  not  meet  the  axes  at  all  ? 

32.  Show  that  the  circle  o;^  +  y^lOor-lOy  +  25  =  0 
touches  the  axes  and  lies  entirely  in  the  second  quadrant. 
Write  the  equation  so  that  it  shall  represent  the  same  circle 
touching  the  axes  and  lying  in  the  third  quadrant. 

33.  In  what  points  does  the  straight  line  3a:  +  y  =  25  cut 
the  circle  a:^  +  y"  =  65  ? 

34.  Find  the  points  common  to  the  loci  x^  ^y^  =  25  and 

y  =  2a:  — 4. 

35.  The  equation  of  a  chord  of  the  circle  o;'^  +  y^  =  4  is 
y  =  2a;  +  ll.     Find  its  length. 

X        V 

36.  The  equation  of  a  chord  is  -  +  V  =1 ;  that,  of  the  circle 
is  x"^  -{-y"^  =  r^     Find  the  length  of  the  chord. 

37.  Find  the  equation  of  a  line  passing  through  the  centre 
oi  x"^ -^ y'^  —  ^ X  —  ^y  =  —  21  and  perpendicular  to  x-\-2y=  4. 

38.  Find  the  equation  of  that  chord  of  the  circle  x'^-\-y~=loO 
which  passes  through  the  point  for  which  the  abscissa  is  9 
and  the  ordinate  negative,  and  which  is  II  to  the  straight  line 
4a:-5y  — 7=0. 

39.  What  is  the  equation  of  the  chord  of  the  circle 
•r''^  +  ?/^  =  277  which  passes  through  (3,-5)  and  is  bisected 
at  this  point  ? 

40.  Find  the  locus  of  the  centre  of  a  circle  j'^assing  through 
the  points  (oTi, yi)  and  {x^,y^. 

41.  What  is  the  locus  of  the  centres  of  all  the  circles  which 
pass  through  the  points  (5,  3)  and  (—7,  —  6)  ? 


82  ANALYTIC    GEOMETRY. 

Find  tlie  equation  of  a  circle  : 

42.  Passing  through  the  points  (4,  0),  (0,  4),  (6,  4). 

43.  Passing  through  the  points  (0,  0),  (8,  0),  (0,  -  G). 

44.  Passing  through  the  points  (-  G,  -1),  (0,  0),  (0,  -1). 

45.  Passing  through  the  points  (0,  0),  (—8a,  0),  (0,  6 a). 
4G.    Passing  through  the  points  (2,  -  3),  (3,  -  4),  (-  2,  - 1). 

47.  Passing  through  the  points  (1,  2),  (1,  3),  (2,  5). 

48.  Passing  through  (10,  4)  and  (17,  —  3),  and  radius  =  13. 

49.  Passing  through  (3,  G),  and  touching  the  axes. 

50.  Touching  each  axis  at  the  distance  4  from  the  origin. 

51.  Touching  each  axis  at  the  distance  a  from  the  orimn. 

52.  Passing  through  the   origin,  and   cutting  the  lengths 
a,  h  from  the  axes. 

53.  Passing  through   (5,  G),   and  having  its  centre  at  the 
intersection  of  the  lines  y  --=^lx  —  3,  4?/  —  ox  =  13. 

54.  Passing  through  (10,  9)  and  (5,  2  — 3VG),  and  having 
its  centre  in  the  line  3a:— 2?/  — 17=0. 

55.  Passing  through  the  origin,  and  cutting  equal  lengths 
a  from  the  lines  :r  =  ?/,  a;  +  y  =  0. 

5G.    Circumscribincr  the  triano;le  whose  sides  are  the  lines 

X        II 

V  =  0,  y=-inx-\-h,  — |-t  =  1- 
^  -^  a      b 

57.  Having  for  diameter  the  line  joining  (0,  0)  and  (:ri,7/,). 

58.  Having  for  diameter  the  line  joining  {x^,  y^)  and  {x.^,  ?/■.). 

59.  Having  for  diameter  the  line  joining  the  points  where 


2  _  9 


rx. 


y  =  mx  meets  x^-^-y 

GO.    Having  for  diameter  the  common  chord  of  the  circles 
x^  -\-y'^  =  9'2  ^^^  ^^  _  ^y  _|_  ^2  ^  ^.2^ 


THE    CIRCLE. 


83 


Tangents  and  Normals. 

71.  Let  QPQ  (Fig.  27)  represent  any  curve.  If  the  secant 
QPB  be  turned  about  the  point  P  until  the  point  Q  approaches 
indefinitely  near  to  P,  then  the  ultimate  position,  TT',  of  the 
secant  is  called  the  Tangent  to  the  curve  at  P. 


The  tangent  TT^  is  said  to  touch  the  curve  at  P,  and  the 
point  P  is  called  the  Point  of  Contact. 

The  straight  line  PN  drawn  from  P,  perjoendicular  to 
the  tangent  TT\  is  called  the  Normal  to  the  curve  at  P. 

Let  the  curve  be  referred  to  the  axes  OX,  0  Y,  and  let  M 
be  the  foot  of  the  ordinate  of  the  point  P.  Let  also  the  tan- 
gent and  the  normal  at  P  meet  the  axis  of  x  in  the  points 
P,  JSf,  respectively.  Then  MT\^  called  the  Sabtangent  for  the 
2:)oint  P,  and  JiiYis  called  the  Subnormal. 

72.  To  find  the  equation  of  a  tangent  to  the  circle  x^-\-y'^--r'^, 
at  the  point  of  contact  {x^,  3/1). 

Let  P  (Fig.  28)  be  the  point  ix^,  y^),  and  Q  any  other  point 
{x2,  y-i)  of  the  circle.     Then  the  equation  of  the  line  PQ  is 

y-yi_y2-yi 


X—  Xy 


(1) 


84  ANALYTIC    GEOMETRY. 

If  now  we  make  this  secant  become  a  tangent,  by  turning  it 
about  P  till  P  coincides  with  P,  then  x^  =  x^,  y-i  =  2/i,  and  the 

fraction  -- — —  assumes  the  indeterminate  form  7:- 
x-i  —  Xi  U 

But  we  have  not  yet  introduced  the  conditions  that  P  and 

Q  lie  in  the  circle. 

These  conditions  are 

x,'+?/,'  =  r' 

Subtracting,  (a^/  -  x,^)  +  (y/  -  y,^)  =  0. 

Factoring,  (x^  —  x^  (.r,  +  x^  -|-  (y^  - y^  (y,  +  y^  =  0. 

Whence,  by  transposition  and  division,  we  have 

3/2  —  yi  ^  _  --^2  +  ^1 
X2  —  Xy  3/2  +  2/1 

And  by  substitution  in  (1)  the  equation  of  the  secant 
becomes  y  _  y^  ^      ^.^  _|_  ^.^ 

x-^i^    y-i  +  yi 

Now  let  Q  coincide  with  P,  or  X2  =  ^1,  y^  =  3/1 ;  the  secant 
becomes  a  tangent  at  P,  and  the  equation  becomes 

y  —  yi  ^  _  ^^^ 
x  —  x^yi 

or  XiX  +  yiy  =  x\^  +  3/1'. 

And,  since  a^i^  +  3/1^  =  r^  we  obtain 

ix^i^  +  yi2/  =  :>'-,  [18] 

an  equation  easily  remembered  from  its  symmetry  and  because 
it  may  be  formed  from  o;^  +  3/^  =  r^  by  merely  changing  x^  to 
XiX  and  y^  to  3/13/. 

Note.  The  above  method  of  obtaining  the  equation  of  the  tangent 
to  a  circle  is  appUcable  to  any  curve  whatever.  It  is  sometimes  called 
the  secant  method. 


THE   CIRCLE.  85 

73.  To  find  the  equation  of  the  normal  through  {xx,  y^). 
The  slope  of  the  taiment  is ^• 

Therefore  that  of  the  normal  will  be  —  (§  48). 

Xi 

Hence  the  equation  of  the  normal  is  (§  50) 

0-1 

which  reduces  to  the  form 

y^x  —  x^y  =  O.  [19] 

Therefore  the  normal  passes  through  the  centre. 
\Ve  may  also  obtain  the  same  equation  by  proceeding  as 
in  §  51. 

74.  To  find  the  equations  of  the  tangent  and  normal  to  the 
circle  (x  —  af  +  (y  —  b)-  =  ?-^  at  the  point  of  contact  {x^,  ?/i). 

We  proceed  as  in  §  72,  only  now  the  equations  of  condition 
which  place  (xx,  y^  and  {x-i,  3/2)  011  the  circle  are 

{x,-ay^{y,~hy^r\ 
{x^  -  aj  +  (2/2  -  ly  =  r\ 

After  subtracting  and  factoring,  we  have 

{x.i-x,)  (x,+o:i-2a)  +  (y2-yi)  (y2+yi-25)  =  0, 

whence     .  Vi      yi ^2  4"  ^1      ^  <^ 

x^  —  Xx~      y.,  +  yx  —  '2.b 

Hence  the  equation  of  a  secant  through  {x^,  yi)  and  {x^,  y^)  is 

y-yi  ^  _  ^'2  +  ^1  — 2  a 
x-xi        y-i  +  yi  —  2'5 

Making  X2  =  o:i,  and  3/2  =  yi,  and  reducing,  we  obtain 

(X,  -a)(x-  a)  +  (2/1  -  6)  (y  -  6)  =  r\  [20] 


86  ANALYTIC    GEOMETRY. 

Equation  [20]  may  be  immediately  formed  from  [18]  by 
affixing  —  a  to  the  x  factors  and  —  6  to  the  y  factors,  on  the 
left-hand  side.' 

By  proceeding  as  in  §  73,  we  obtain  for  the  equation  of 
the  normal 

{y^-b){x-x,)-{a^,-a){y-y,)  =  0.  [21] 


75.    To  ji ad  the  condition  that  the  straight  line  y-=inix-\-c 
shcdl  touch  the  circle  x^  +  2/"^  =  ^'^• 

I.    If  the  line  touch  the  circle,  it  is  evident  that  the  perpen- 
dicular from  the  origin  to  the  line  must  be  equal  the  radius  r 

of  the  circle.     The  length  of  this  perpendicular  is 


Vl  +  m'^ 
(§  55).     Therefore  the  required  condition  is  expressed  by  the 

equation 

c^  =  r\l  +  m'). 

II.    By  eliminating  y  from  the  equations 
y  —  mx  +  c,     x^  -\-y^  =  f^, 
we  obtain  the  quadratic  in  x, 

(1  -}-  r)t'^)x'  +  2  mcx  =  7-"^  —  c^^ 
the  two  roots  of  which  are 

—  _     '^^^'^  Vr'(l-f-  '^>^'0  —  <?' 

^^       1  +  ni'  ~  Y^ni' 

If  these  roots  are  real,  the  line  will  cid  the  circle  ;  if  they 
are  equal,  it  will  touch  the  circle  ;  if  they  are  imaginary,  it 
will  not  meet  the  circle  at  all. 

The  roots  will  be  equal  if  ^r\\-\-w})  —  c^  =  0  ;  that  is,  if 
c-  =  9-\l  +  m"^),  a  result  agreeing  with  that  previously  obtained. 

If  in  the  equation  y  =  mx  +  c  we-  substitute  for  c  the  value. 
rVl  +  'm-'*,  we  obtain  the  equation  to  the  tangent  of  a  circle  in 
the  useful  form  .       /r—. :t  rooi 


THE    CIRCLE.  87 

This  equation,  if  we  regard  ??i  as  an  arbitrary  constant, 
represents  all  possible  tangents  to  the  circle  x^  +  if  =  r^ 

Note  1.  Method  II.  is  applicable  to  any  curve,  and  agrees  with  the 
definition  of  a  tangent  given  in  ^  71. 

Note  2.  In  problems  on  tangents  the  learner  should  consider  whether 
the  co-ordinates  of  the  point  of  contact  are  involved.  If  they  are,  he 
should  use  equation  [18] ;  if  they  are  not,  then  in  general  it  is.  better  to 
use  equation  [20]. 

Ex.  22. 

1.  Explain  the  meaning  of  the  double  sign  in  equation  [22], 

2.  Deduce  the  equations  of  the  tangent  and  normal  to  the 
circle  x^A^if  =  r^^  assuming  that  the  normal  passes  through 
the  centre. 

3.  Find  the  equations  of  the  tangent  and  the  normal  pass- 
ing through  the  point  (4,  6)  of  the  circle  .r'"^ -|- y^  =  52.  Also 
the  lengths  of  tangtnt,  normal,  subtangent,  subnormal,  and 
the  portion  of  the  tangent  contained  between  the  axes. 

4.  A  straight  line  touches  the  circle  x^ -\- y- ^=^  r"^  in  the 
point  (a-'i,  ?/,).  Find  the  lengths  of  the  subtangent,  the  sub- 
normal, and  the  portion  of  the  line  contained  between  the 
axes. 

5.  What  is  the  equation  of  a  tangent  to  the  circle 
a;2  _j_  ^2  _  o^Q  ^^  ^i^g  point  whose  abscissa  is  9  and  ordinate 
negative  ? 

6.  Find  the  equations  of  tangents  to  .^■^ +  2/^—10  ^t  the 
points  whose  common  abscissa  =1. 

7.  Tangents  are  drawn  through  the  points  of  the  circle 
a;'-[-y2  — 25  which  have  abscissas  numerically  equal  to  3.  Prove 
that  these  tangents  enclose  a  rhombus,  and  find  its  area. 

8.  The  subtangent  for  a  certain  pjoint  of  a  circle  is  5|- ;  the 
subnormal  is  3.     What  is  the  equation  of  the  circle  ? 


88  ANALYTIC    GEOMETRY. 

Find  the  equation  of  a  straight  line 

9.    Touching  .r^-f-y^  =  232  at  the  point  whose  abscissa  =14. 

10.  Touching  {x  -  2)'  +  (y  —  3)'  =  10  at  the  point  (5,  4). 

11.  Touching  a:^  + y^  — 3:r  — 4?/  =  0  at  the  origin. 

12.  Touching  x'^-\-y'^  —  \^x  —  4?/— 5  =  0  at  the  point  whose 
abscissa  is  equal  to  10. 

What  is  the  equation  of  a  straight  line  touching  the  circle 
x^  -\-y-  ^  r^  and  also 

13.  Passing  through  the  point  of  contact  (r,  0)  ? 

14.  Parallel  to  the  line  Ax  +  ^y  +  (7=  0  ? 

15.  Perpendicular  to  the  line  Ax  -{-By  +  C=  0  ? 

16.  Making  the  angle  45°  with  the  axis  of  :r? 

17.  Passing  through  the  exterior  point  (A,  0)  ? 

18.  Cutting  off  a  triangle  of  area  F'  from  the  axes  ? 

19.  Find  the  equations  of  the  tangents  drawn  from  the 
point  (10,  5)  to  the  circle  x""  +  y'  =  100. 

20.  Find  the  equations  of  tangents  to  the  circle  x"^  +  if 
-\-l0x  —  Q>y-2  =  0  and  II  to  the  line  y  =  2x-1. 

21.  Find  the  lengths  of  subtangent  and  subnormal  in  the 
circle  a:^  +  ?/^  — 14a;  — 4?/  — 5  for  the  point  (10,  9). 

22.  What  is  the  equation  of  the  circle  (centre  at  origin) 
which  is  touched  by  the  straight  line  :?;  cos  a -f- y  sin  a  ^=j(:>  ? 
What  are  the  co-ordinates  of  the  point  of  contact  ? 

23.  When  will  the  line  Ax  ^  By -{-0=0  touch  the  circle 
x'^y''  =  r'  ?  the  circle  {x  ~  a)'  +  (y  —  hf  =  r'  ? 

24.  Find  the  equation  of  a  straight  line  touching  x"^ -[- y^ 
=  ax  -\-hy  and  passing  through  the  origin. 


THE    CIRCLE.  89 

Prove  that  the  following  circles  and  straight  line  touch, 
and  find  the  jiuints  of  contact  in  each  case  : 

25.  X-  +  /  +  ax  -\-hij  =  0  and  ax-^hy-{-  a^  +  Z.^  ^  0. 

26.  a--  +  y'  — 2a.T-2Z)?/  +  Z>'  =  0  and  x^2a. 

27.  x"^  -\- if  =  ax -\~  hy  and  ax  —  hy-{-h'^  =  0. 

28.  AVhat  is  the  equation  of  the  circle  (centre  at  origin) 
"^'hich  touches  the  line  y  =  3a:  —  5  ? 

29.  AVhat  must  he  the  value  of  m  in  order  that  the  line 
y  =  mx  +  10  may  touch  the  circle  x"^  -{-y"^  ^  100  ?  Show  that 
we  get  the  same  answer  for  the  line  ?/  =  inx  — 10,  and  explain 
the  reason. 

30.  Determine  the  value  of  cin  order  that  the  line  3a7  — 4y 
-f- c  =:  0  may  touch  the  circle  x^-^-y"^  —  ^x-^-l'iy  —  44=0. 
Explain  the  double  answer. 

31.  What  is  the  equation  of  the  circle  having  for  centre 
the  point  (5,  3)  and  touching  the  line  3.r  -{-  2?/  — 10  =  0  ? 

32.  What  is  the  equation  of  a  circle  whose  radius  =  10, 
and  which  touches  the  line  4a;-{-3y  — 70  =  0  in  the  point 
(10,10)? 

33.  About  the  point  (5,  9)  a  circle  touching  the  line 
4:r  +  3?/  +  3  =  0  in  the  point  (-3,3)  is  described.  What 
is  its  equation  ? 

34.  Under  what  condition  will  the  line  -  +  ^  =  1  touch 
the  circle  x^  -f  y^  _  ^.2  ^ 

35.  What  is  the  equation  of  the  circle  inscribed  in  the 
triangle  whose  sides  are 

.  =  0.  y  =  o,  -^+1=1? 


90  ANALYTIC    GEOMETRY. 

3G.  Two  circles  touch  each  other  when  the  distance  between 
their  centres  is  equal  to  the  sum  or  the  difference  of  their 
radii.     Prove  that  the  circles 

x-^  -\-f  =  (T  +  ay,  {x  -  af  i-i/  =  r' 

touch  each  other,  and  find  the  equation  of  the  common  tangent. 

37.  Two  circles  touch  each  other  when  the  length  of  their 
common  chord  =  0.     Find  the  length  of  the  common  chord  of 

(x  -  ay  +  (y  -  by  =  r',       (x  -  by  +  (y  -  ay  -  r' , 

and  hence  prove  that  the  two  circles  touch  each  other  when 
(a-by  =  2r\ 

Ex.  23.      (Review.) 

Find  the  radii  and  centres  of  the  following  circles: 

1.  Sx'-6x  +  32/-i-9y-V2  =  0. 

2.  7x'  +  S7/-4:7/-{l-2xy=^0. 

3.  y(y~b)  =  x(S-x). 


4.  ■\/l+a;'(x''  +  7/)^2b(x  +  a7/). 

Findxhe  equation  of  a  circle  : 

6.  Centre  (0,  0),  radius  =  9. 

6.  Centre  (7,  0),  radius  =  3. 

7.  Centre  (—  2,  5),  radius  =  10. 

8.  Centre  (3  a,  4a),  radius  =  5a. 

9.  Centre  (b  -^  c,  b  —  c),  radius  —  c. 

10.  Passing  through  (a,  0),  (0,  b),  (2a,  2b). 

11.  Passing  through  (0,  0),  (0,  12),  (5,  0). 

12.  Passing  through  (10,  9),  (4,-5),  (0,  5). 

13.  Touching  each  axis  at  the  distance  —7  from  the  origin. 


THE    CIRCLE.  91 

14.  Touching  both  axes,  and  radius  =  r. 

15.  Centre  (a,  a),  and  cutting  chord  =  h  from  each  axis. 
IG.    Passing  through  (0,  0),  and  touching  7/  =  2x-{-S. 

17.    Passing  through  (1,  —  3),  and  touching  2x  —  i/—  4  =  0. 

IS.  With  its  centre  in  the  line  5.r  — 7y— 8  =  0,  and 
touching  the  Hnes  2a;  — ?/  =  0,  x  —  2i/  —  6  =  0. 

19.  Passing  through  the  origin  and  the  points  common  to 
the  circles  ^^2  ^  ^2  _  g^  _  j^Oy  -  15  =  0, 

x'  +  f  +  2x-{-    4y  +  20  =  0. 

20.  Having  its  centre  in  the  line  5x  —  Si/  —  7  —  0,  and 
passing  through  the  points  common  to  the  same  circles  as  in 
No.  19. 

21.  Touching  the  axis  of  x,  and  passing  through  the  points 
common  to  the  circles 

x-'^y~-j-4:x  —  Ui/  -  68  =  0, 
^.2  ^  y2  __  Q^.  _.  22^  _^  30  ==  0. 

22.  Find  the  centre  and  the  radius  of  the  circle  which 
passes  through  (9,  6),  (10,  5),  (3,  -2). 

23.  What  is  the  distance  from  the  centre  of  the  circle 
passing  through  (2,  0),  (8,  0),  (5,  9)  to  the  straight  line  joining 
(0,-11)  and  (-16,1)? 

24.  What  is  the  distance  from  the  centre  of  the  circle 
x'  +  7/-  4:x  4-  8y  =  0  to  the  line  4.i-  -  3//  +  30  =  0  ? 

25.  W^hat  portion  of  the  line  ^  =  5x-i-2  is  contained  within 
the  circle  x'  -^  f-  -  ISx  -  4:7/  —  9  =  0? 

26.  Through  that  point  of  the  circle  x^^if  =  25  for  which 
the  abscissa  ^=  4  and  the  ordinate  is  negative,  a  straight  line 
parallel  to  y  =  Zx  —  ^  is  drawn.  Find  the  length  of  the  inter- 
cepted chord. 


92  ANALYTIC    GEOMETRY. 

27.  Through  the  point  (xi,i/i),  within  the  circle  x'^ -^  7/^  =  7'^ , 
a  chord  is  drawn  so  as  to  be  bisected  at  this  point.  What  is 
its  equation  ? 

28.  What  relation  must  exist  among  the  coefficients  of  the 
equation  Aix'  +  y')  +  Bx  +1^1/+  C  =  0. 

(i.)  in  order  that  the  circle  may  touch  the  axis  oi  x? 
(ii.)  in  order  that  the  circle  may  touch  the  axis  of  3/  ? 
(iii.)  in  order  that  the  circle  may  touch  both  axes  ? 

29.  Under  what  condition  will  the  straight  line  2/  =  7nx  -f  c 
touch  the  circle  x'  -\-  y  =^2  rx  ? 

30.  What  must  be  the  value  of  Iz  in  order  that  the  line 
3  a:  +  4?/  =  Z;  may  touch  the  circle  y'^  =  IC-r  —  rr'  ? 

31.  Find  the  equation  of  the  circle  which  passes  through 
the  origin  and  cuts  equal  lengths  a  from  the  lines  ^  =  y, 
a;  4-  y  =  0. 

32.  Find  the  ecjuations  of  the  four  circles  whose  common 
radius  -=  V2ct,  and  which  cut  chords  from  each  axis  equal 
to  2  a. 

33.  Find  the  equation  of  the  circle  whose  diameter  is  the 
common  chord  of  the  circles  x^  -\-y^  =^  r^,  {x  —  df  -\-  y"^  =  r"^. 

Find  the  equation  of  the  straight  line 

34.  Passing  through  (0,  0)  and  the  centre  of  the  circle 

^,2  _|_  y2  ^  a{x  +  y). 

35.  Passing  through  the  centres  of  the  circles 

a;-  -j-  y-  =  25  and  .r^  +  y^  +  6 a;  —  Sy  =  0. 

36.  Passing  through  (0,  0)  and  touching  the  circle 

^•'  +  y^  -  6 5;  -  12y  4-  41  =  0. 

37.  Parallel  to  .r-f  V3(y-12)  =  0  and  touching  x''^y''=lOO. 


THE    CIRCLE.  93 

38.  Passing  through  the  points  common  to  the  circles 

x'-\-y--    2x-    47/-    20-0, 
:c-  +  y'-14.x"-lG?/+  100  =  0. 

39.  Prove  that  the  common  chord  of  the  circles  in  No.  38 
is  perpendicular  to  the  straight  line  joining  their  centres. 

40.  Find  the  area  of  the  triangle  formed  by  radii  of  the 
circle  .r^-f2/"  =  169  drawn  to  the  points  whose  abscissas  are 
—  12  and  +7  and  ordinates  positive,  and  the  chord  passing 
through  the  same  two  points. 

41.  Prove  that  an  ancrle  inscribed  in  a  semicircle  is  a  rit^ht 

o  o 

angle. 

42.  Prove  that  the  radius  of  a  circle  drawn  perpendicular 
to  a  chord  bisects  the  chord. 

43.  Find  the  inclination  to  the  axis  oi  x  of  the  line  joining 
the  centres  of  the  circles  x"^  +  2a;  -f  y^  =  0,  x^  +  2y  +  2/"  =  0. 

44.  Determine  the  point  from  which  tangents  drawn  to  the 
circles  ^^^' +  v/ -    2x-    6?/+    6  =  0, 

1^  -f  y2  _  ooy  _  20 ^^.  +  52  =  0, 

will  each  be  equal  to  4V6. 

45.  Find  the  equations  of  the  circles  which  touch  the 
straight  lines  6a:  +  7?/  +  9  =  0  and  72-  +  G?/  +  3  =  0,  and 
the  latter  line  in  the  point  (3,  —4). 

Obtain  and  discuss  the  ecpiations  of  the  following  loci : 

46.  Locus  of  the  centres  of  a  circle  having  the  radius  r  and 
passing  through  the  point  (.^i,  y^). 

47.  Locus  of  the  centre  of  a  circle  having  the  radius  r'  and 
touching  the  circle  {x  —  a)^  -\- (y  —  hy  =  r^ 

48.  Locus  of  all  points  from  which  tangents  drawn  to  the 
circle  {x  —  a)-  +  (y  —  by  =  r^  have  a  given  length  t. 


94  ANALYTIC    GEOMETRY. 

49.  Locus  of  the  middle  point  of  a  chord  drawn  through  a 
fixed  point  J.  of  a  given  circle. 

50.  Locus  of  the  point  31  which  divides  the  chord  AC, 
drawn  through  the  fixed  point  ^  of  a  given  circle,  in  a  given 
ratio  AM:  MC  =  m  :  n. 

5L  Locus  of  a  point  whose  distances  from  two  fixed  points, 
A,  B,  are  in  a  constant  ratio  m :  n. 

52.  Locus  of  a  point,  the  sum  of  the  squares  of  whose  dis- 
tances from  two  fixed  points,  A  and  B,  is  constant,  and  equal 
to  Tc. 

53.  Locus  of  a  point,  the  difference  of  the  squares  of  whose 
distances  from  two  fixed  points,  ^,  ^,  is  constant  and  equal 
to  Ir. 

54.  Locus  of  the  middle  point  of  a  line  of  constant  length 
d  which  moves  so  that  its  ends  always  touch  two  fixed  per- 
pendicular lines. 

55.  Locus  of  the  vertex  of  a  triangle  whose  base  is  fixed 
and  of  constant  length,  and  the  angle  at  the  vertex  is  also 
constant. 

56.  One  side,  AB,  of  a  triangle  is  constant  in  length  and 
fixed  in  position  ;  another  side,  AC,  is  constant  in  length  but 
revolves  about  the  point  A,  Find  the  locus  of  the  middle 
point  of  the  third  side,  BC. 

57.  Find  the  locus  of  the  intersections  of  tangents  at  the 
extremities  of  a  chord  whose  length  is  constant. 

58.  Given  the  equation  of  a  circle  x^  +  lf  =  r"^-  If  its  radii 
are  produced,  each  by  a  length  equal  to  the  abscissa  of  the 
point  where  it  meets  the  circle,  find  the  locus  of  the  extremi- 
ties of  the  radii  produced. 


THE    CIRCLE. 


95 


SUPPLEMENTARY   PROPOSITIONS. 

76.    To  find  the  locus  of  the  middle  points  of  a  systenn  of 
parallel  chords  in  the  circle  x'  -f  y'^  ^=  r"^. 


Fig.  29. 

Let  the  equation  of  any  one  of  the  chords  (Fig.  29)  be 
y^=mX'\-c,  and  let  it  meet  the  circle  in  the  points  (xi,  y^  and 
(^'2, 2/2). 

Then  (§§  37  and  72)  m  =  -  ^^^^  (1) 


2/1  +  ^2 

Let  (.r,  y)  be  the  middle  point  of  the  chord  ;  then  2x=Xi-]-X2, 
^y^^yi~^y-i  (§  S))  ^^^^  %  substitution  we  have 

(2) 


X 

y 


a  relation  which  evidently  holds  true  for  the  middle  points  of 
all  the  chords.     Therefore  (2)  is  the  equation  of  the  locus. 
If  we  write  (2)  in  the  form 

we  see  that  the  locus  is  a  straight  line  passing  through  the 
centre,  and  perpendicular  to  the  chords  (§  48). 

The  locus  of  the  middle  points  of  a  system  of  parallel 
chords  is  called  a  Diameter  of  the  circle;  and  the  chords 
which  it  bisects  are  called  the  Ordinates  of  the  diameter. 


96 


ANALYTIC    GEOMETRY. 


77.  Two  tangents  can  he  drawn  to  a  ch'cle  from  any  point ; 
and  these  tangents  will  he  real,  coincident,  or  imaginary ,  ac- 
cording as  the  point  is  outside,  on,  or  inside  the  circle,  respec- 
tively. 


Fig.   30. 

Let  the  equation  of  the  circle  be 
x'  +  f  =  r\ 
Let  {xi,  3/1)  be  the  point  of  contact  of  a  tangent,  (A,  h)  any 
other  point  in  the  tangent.      Then  {h,  k)   must  satisfy   the 
equation  of  the  tangent;  therefore 

xji  +  yjc  =  r\  (1) 

Also,  since  (xi,  y^  is  on  the  circle, 


Xi  +  3/1 


Eliminating  3/1,  we  have 


^V  + 


li^Y^  ,.2^ 


(2) 


(3) 


Since  equation  (3)  is  a  quadratic  equation,  there  are  two 
points  the  tangents  at  which  pass  through  (A,  k).  Solving 
(3),  we  obtain  _  ^.^  ^,^,.^i^^^_^ 

and  we  see  that  the  values  of  Xi  are  o^eal,  coincident,  or  imagi- 
nary, according  as  A"  -|-  F  is  greater  than,  eciual  to,  or  less 
than  r^ ;  that  is  to  say,  according  as  (A,  Ic)  is  outside,  on,  or 
inside  the  circle. 


THE   CIRCLE.  97 

78.  Tangents  arc  drawn  to  the  circle  x'  +  y^  ~  '^'^  from  any 
point  (A,  I:)  ;  to  find  the  equation  of  the  straight  line  joining 
the  two  points  of  contact. 

Let  {i\,  ijx),  (x.,,  7/2)  be  the  points  of  contact ;  then  the  equa- 
tions of  the  tangents  are  (§  72) 

Since  both  tangents  pass  through  (A,  Tc),  both  these  equa- 
tions are  satisfied  by  tlie  co-ordinates  A,  Iz ;  therefore 

hx,-{-hj,=^r\  (1) 

hx,-^hj,  =  r\  (2) 

From  equations  (1)  and  (2)  we  see  that  the  two  points 
G"^'!,  yi)»  (^'2,  y-i)  both  satisfy  the  equation 

hx-\-lcy  =  r\  (3) 

which,  as  its  form  shows,  represents  some  straight  line.  There- 
fore equation  (3)  is  the  equation  of  the  straight  line  passing 
through  {xx,y^  and  {x^.y-^;  in  other  words,  .the  equation 
required. 

The  line  represented  by  equation  (3)  is  a  real  line,  whether 
(A,  Ic)  be  outside  or  inside  the  circle. 

If  the  point  (A,  Ic)  be  outside  the  circle,  this  line  is  called 
the  Chord  of  Contact  of  the  two  real  tangents  drawn  from  (A,  U). 

If  the  point  (A,  Ic)  be  inside  the  circle,  the  points  of  contact 
and  the  tangents  are  imaginary,  and  we  have  a  real  line 
joining  two  imaginary  points. 

79.  The  straight  line  passing  through  the  points  of  contact 
of  the  tangents  (real  or  imaginary)  which  can  be  drawn  from 
any  point  to  a  circle  is  called  the  Polar  of  that  point  with 
respect  to  the  circle  ;  and  the  point  is  called  the  Pole  of  that 
straight  line  with  respect  to  the  circle. 


98  ANALYTIC    GEOMETRY. 

80.  If  the  polar  of  a  'point  P  pass  through  Q,  then  the 
polar  of  Q  will  pass  through  JP. 

Let  P  be  the  point  (h,  Jc),  Q  the  point  (h',  Jc'),  and  let  the 
equation  of  the  circle  be  x'  +  y^  =  '''^ 

Then  the  equations  of  the  polars  of  P  and  Q  are 

hx  +  hj  =  7•^  (1) 

h'x  +  k'7/  =  r\  (2) 

If  Q  be  on  the  polar  of  P,  its  co-ordinates  must  satisfy 
equation  (1) ;  therefore 

hh'-\-kk'  =  r\ 

But  this  is  also  the  condition  that  P  shall  be  on  the  line 
represented  by  (2)  ;  that  is,  on  the  polar  of  Q.  Therefore  P 
is  on  the  polar  of  Q. 

81.  Jf  a  straight  line  revolve  about  a  fixed  point  Q,  and  P 
is  the  pole  of  that  line,  the  locus  of  P  is  the  pjolar  of  Q. 

For,  since  Q  is  on  the  polar  of  P,  the  point  P  must  always 
be  on  the  polar  of  Q  (§  80). 

82.  If  a  pjoint  Q  move  along  a  fixed  straight  line,  and  P  is 
the  pole  of  that  line,  then  the  polar  of  Q  will  revolve  about  P. 

For,  by  hypothesis,  the  polar  of  P  passes  through  Q  (§  80). 

83.  The  polar  of  a  point  luith  respect  to  a  circle  is  perpen- 
dicular to  the  line  joining  the  point  to  the  centre  of  the  circle. 

Let  the  equation  of  the  circle  be 

and  let  P  be  any  point  (h,  k).     Then   the  equation   of  the 
polar  of  Pis  hx-{-hj  =  r\  (1) 

And  the  equation  of  the  line  joining  P  to  the  centre  0  of 
the  circle  is  l:x-hy  =  0.  (2) 


THE    CIRCLE. 


99 


The  form  of  equations  (1)  and  (2)  shows  that  the  lines 
which  they  represent  are  perpendicular  (§  51). 

Figs.  31  and  32  illustrate  the  relations  of  poles  and  polars 
which  have  been  established  in  §§  80-83. 


100 


ANALYTIC    GEOMETRY. 


84.    To  find  a  geometrical  construction  for  the  polar  of  a 
point  with  respect  to  a  circle. 


Fig.  33. 


Fig.    34. 


If  the  notation  of  §  83  be  retained,  and  OQ  (Figs.  33  and  34) 
be  the  perpendicular  from  0  to  the  pokir  of  P,  then  (§  55) 

0Q  = 


Also 

Therefore 


V/r  +  li' 
OFX  OQ  =  r\ 


Hence  we  have  the  following  construction  : 
Join  OF,  and  let  it  cut  the  circle  in  A  ;  take  Q  in  the  line 
OF,  such  that  Qp ,  qj,  ^  qA  :  OQ, 

and  draw  through  Q  a  line  perpendicular  to  OF. 

85.    To  find  the  length  of  the  tangent  drawn  from  any  point 
(h,  k)  to  the  circle   (^,  _  ^y  -f  (y  —  hf  -  r'  =  0.  (1) 

Let  F  (Fig.  35)  be  the  point  {h,  h),  Q  the  point  of  contact, 
Cthe  centre  of  the  circle  ;  then,  since  FQCis  a  right  angle, 

FQ'  =  FC'-QC\ 

Now  FC':=(h-ay--}-(I:-by\    QC'  =  r\ 


Therefore 


FQ'  =  (h  -  ay  -i-  (Ic  -  bf  -r 


THE    CIRCLE. 


101 


Hence  PQ-  is  found,  by  simply  substituting  tlie  co-ordinates 
of  F  in  the  left-hand  member  of  equation  (1). 

If  for  brevity  we  write  S  instead  of  (x  —  of  -f  (y  —  hy  —  o'"^, 
then  the  equation  8—  0  will  represent  the  general  equation 
of  the  circle  after  division  by  the  common  coefficient  of  :r 
and  y^,  and  we  may  state  the  above  result  as  follows  : 

If  S^=0  be  the  equation  of  a  circle,  and  the  co-ordinates  of 
any  point  he  substituted  for  x  ayid  y  in  S,  the  result  luill  he 
equal  to  the  square  of  the  length  of  the  tangent  drawn  from  the 
2:)oint  to  the  circle. 

If  the  point  is  inside  the  circle,  the  square  is  negative,  and 
the  length  of  the  tangent  imaginary. 

86.  Two  circles  intersect  each  other ;  to  find  the  equation  of 
the  straight  line  passing  through  the  points  of  intersection. 


X 


Fi2.   35. 


Fig.  36. 


Let  the  equations  of  the  circles  be 

(x-a)'  +  (y-J)'-r^=0,  (1) 

,     {x-a'f  +  (,j-h'y-r"  =  0.  (2) 

Subtract  one  of  these  equations  from  the  other;  then 
2 (a-  a')x  +  2 (^-  -  hyj  =  cr -  a"  +b'-  h" -  (r' - r")  =  0. 

This  is  the  equation  required  ;  for  it  is  the  equation  of 
some  straight  line,  and  its  locus  passes  through  the  intersec- 
tions of  the  loci  represented  by  (1)  and  (2)  (§  59). 


102  ANALYTIC    GEOMETRY. 

If  we  write  Si  and  /S2  for  the  left-hand  members  of  equations 
(1)  and  (2)  respectively,  the  result  may  be  thus  stated : 

Jf  /S^  =z  0,  /S.2  =  0  he  the  equations  of  two  circles,  then  will 
the  equation  S^  —  /S^  =  0,  or  Si  =  /S^  he  the  equation  of  the 
straigJd  line  through  their  points  of  intersection. 

Although  the  two  circles  Si  ==  0,  /S2  =  0  may  not  cut  each 
other  in  real  points,  the  straight  line  Si  —  S-^  will  always  be 
real,  provided  the  constants  in  it  are  real ;  so  that  we  have 
a  real  straight  line  passing  through  imaginary  points. 

But  another  meaning  may  be  given  to  the  equation  Si  —  S.^. 

For  if  (x,  y)  denote  any  point  in  the  line  Si  —  /Sj,  then  xSi 
is  equal  to  the  square  of  the  tangent  from  {x,  y)  to  the  circle 
Si  =  0,  and  S-i  is  equal  to  the  square  of  the  tangent  from 
{x,  ?/)  to  the  circle  S.  =  0  (§  85). 

Hence  the  tangents  drawn  to  the  two  circles  from  any 
point  in  the  straight  line  Si  =  S^  are  equal. 

The  straight  line  X  —  S.,  is  called  the  Eadical  Axis  of  the 

O  J.  - 

two  circles  /Si  —  0,  xS'a  =  0. 

It  may  be  defined  either  as  the  straight  line  passing  tJtrough 
the  ^ooints  of  intersection  (real  or  imo,ginary)  of  the  two  circles, 
or  as  the  locus  of  the  points  from  which  tangents  drawn  to  the 
two  circles  are  equal. 

87.  The  three  radical  axes  of  three  circles,  tahen  in  pairs, 
m.ect  in  a  point. 

Let  /6'=0,  Si^=0,  /Sj  =  0  be  the  equations  of  the  circles, 
when  the  coefficient  of  x^  in  each  is  unity. 

Then  the  equations  of  their  radical  axes,  taken  in  pairs,  are 

S-Si  =  0,     Si-S,  =  0,     S-S,=0. 

The  values  of  x  and  y  that  will  satisfy  any  two  of  these 
equations  will  also  satisfy  the  third.  Therefore  the  third 
axis  passes  through  the  point  of  intersection  of  the  other 
two  axes.  The  point  of  intersection  of  the  three  radical  axes 
is  called  the  Eadical  Centre  of  the  three  circles. 


THE    CIRCLE.  103 

Ex.   24. 

1.  What  is  tlie  equation  of  the  diameter  of  the  circle 
x'^-}-i/  =  20  which  bisects  chords  parallel  to  the  line  6:r  +  7?/ 
+  8  =  0? 

2.  What  is  the  equation  of  the  diameter  of  the  circle  which 
bisects  all  chords  whose  inclination  to  the  axis  of  x  is  135°  ? 

3.  Prove  that  the  tangents  at  the  extremities  of  a  diameter 
are  parallel. 

4.  Write  the  equations  of  the  chords  of  contact  in  the  circle 
^^  +  y"  —  ''^  foi-"  tangents  drawn  from  the  following  points : 
(r,r),  (2r,  3r),  (a  +  b,a-b). 

6.  From  the  point  (13,  2)  tangents  are  drawn  to  the  circle 
^'  +  y^  =  49  ;  what  is  the  equations  of  the  chord  of  contact  ? 

6.  What  line  is  represented  by  the  equation  hx  -{-  J:y  =  r^ 
w^hen  (/i,  Ic)  is  in  the  circle  ? 

7.  Write  the  equations  of  the  polars  of  the  following  j^oints 
with  respect  to  the  circle  o;^  +  ?/^  =  4  : 

(i.)(2,3).        (ii.)(3,-l).        (iii.)  (1,-1). 

8.  Find  the  poles  of  the  following  lines  with  respect  to  the 
circle  x^  +  ?/^  =  35  : 

(i.)  43;  +  6y=--7.        (ii.)  3a;-2?/=5.        (iii.)  ax^ly^X. 

9.  Find  the  pole  of  3.'r-f  4?/  =  7  with  respect  to  the  circle 
X'^y'^  14. 

10.  Find  the  pole  of  Ax^By^C^=^  with  respect  to  the 
circle  x-  -\-  y^  =  r\ 

11.  Find  the  co-ordinates  of  the  points  w^here  the  line  x==4: 
cuts  the  circle  X'  +  v/-  =  4  ;  also  find  the  equations  of  the  tan- 
gents at  those  points,  and  show  that  they  intersect  in  the 
point  (1,  0). 


104  ANALYTIC    GEOMETRY. 

12.  If  the  jiolars  of  two  points  P,  Q  meet  in  R,  then  B  is 
the  pole  of  the  line  PQ. 

13.  If  the  polar  of  (A,  h)  with  respect  to  the  circle  x'^-{-y''=r^ 
touch  the  circle  x^ -\- y^  —  'Zo-x,  then  k'  —  2 rh  =  r\ 

14.  If  the  pole  lie  in  the  circle  a;'-f3/^  =  4c^,  then  the 
polar  will  touch  the  circle  4(:t^^  +  y")  =  <?^- 

15.  Find  the  polar  of  the  centre  of  the  circle  rr-  +  y^  =  r^ 
Trace  the  changes  in  the  position  of  the  polar  as  the  pole  is 
supposed  to  move  from  the  centre  to  an  infinite  distance. 

16.  AVhat  is  the  square  of  the  tangent  dra.wn  from  the 
point  {h,  I:)  to  the  circle  x'  +  y'  =  r'  ? 

17.  Find  the  length  of  the  tangent  drawn  from  (2,  5)  to 
the  circle  x^  -{-  y"^  —  2x  —  oy  —  1=  0. 

Find  the  radical  axis  of  the  circles 

18.  (x  +  5y  +  (y  +  6y  =  9,     (.^^_77  +  (y-H)^  =  16. 

19.  a;^  +  ?/2  +  2.T  +  3?/-7-=0,     x'  +  y- -  2x -y +  1=0. 

20.  X-  +  y"^  +  bx  -\-  by  ~  c  =  0,     ax"^  +  «2/"  +  ct^^  +  ^V  =  ^^ 

21.  Find  the  radical  axis  and  length  of  the  common  chord 
of  the  circles 

.T"  +  y^  +  ax  +  5?/  +  (?  =  0,     x"^  +  yi^  +  bx  -\-ay-\-c  =  0. 

22.  Find  the  radical  centre  of  the  three  circles 

.T^  +  ?/2  +  4a:  +  7=0, 

2x^  +  27/2  +  3:i-  +  5y  +  9  =  0, 

x'-\-f-\-y  =  0. 

23.  Prove  that  the  radical  axis  of  two  circles  is  perpen- 
dicular to  the  straight  line  joining  their  centres. 

24.  Find  a  geometric  construction  for  the  radical  axis  of 
two  circles  which  do  not  meet  each  other  (see  §  87  and  Ex.  23). 


CHAPTER  IV. 

DIFFERENT    SYSTEMS    OF    CO-ORDINATES. 
Oblique  Co-ordinates. 

88.  When  we  define  the  position  of  a  point  or  a  line  by 
reference  to  some  system  of  points  or  lines  regarded  as  fixed 
in  position,  we  are  said  to  employ  a  System  of  Oo-ordinates. 

The  system  of  co-ordinates  which  we  have  thus  far  em- 
ployed is  called  the  Eectangular  System,  because  the  two  fixed 
lines  of  reference  are  perj^endicular  to  each  other.  It  is  the 
system  to  be  preferred  for  most  purposes  on  account  of  its 
simplicity. 

There  are,  however,  two  other  systems  in  use,  of  such  im- 
portance that  we  shall  briefly  describe  and  illustrate  them. 


The  first  of  these  'svstems  differs  from  the  rectangular 
system  simply  in  the  fact  that  the  axes  of  reference  are  not 
perpendicular  to  each  other. 

Let  OX,  OF  (Fig.  37)  be  two  axes  making  an  acute  angle, 
XO  F=  (u,  with  each  other.     The  position  of  the  point  P  is 


106  ANALYTIC    GEOMETRY. 

determined  by  stating  its  distance  from  each  axis,  measured 
along  a  line  parallel  to  the  other  axis. 

If  we  draw  FN  II  to  OX,  and  PM  II  to  0  F,  then  the  co- 
ordinates of  F  are 

NF=OM=x,     MF^y, 


This  system  of  co-ordinates  is  known  as  the  Obhque  System, 
Eectangular   and   oblique   co-ordinates  are    called   Parallel 

Co-ordinates ;   also  Cartesian  Co-ordinates  (from  Descartes,  who 

first  used  them). 

89.  To  find  the  equation  of  the  straight  line  AC,  referred  to 
the  oblique  axes  OX,  0  Y  (Fig.  38),  having  given  the  intercept 
OB  =  b  and  the  angle  XAC^=  y. 

Let  F  be  any  point  {x,  y)  of  the  line.  Draw  BL  \\  to  OX, 
meeting  FM'ui  D.     Then,  by  Trigonometry, 

FD           sin  y                 y  —  b            sin  y 
. =  _ L — ,    or  ^ ■  = f- 

BD      sin  (w  —  y)  X         sin  (w  —  y) 

If  now  we  put  m  =  - — - — "^ — -,  we  obtain  as  the  result  an 
sin  (o)  —  y) 

equation  of  the  same  form  as  [6],  p.  38, 

y  =  nix  -f-  b. 

What  does  the  value  of  m  become  in  this  equation  when 

o>  =  90°  ? 


DIFFERENT    SYSTEMS    OF    CO-ORDINATES. 


107 


90.  Oblique  co-ordinates  are  seldom  used,  because  they 
generally  lead  to  more  complex  formulas  than  rectangular 
ones.  In  certain  cases,  however,  they  may  be  employed  to 
advantage.  An  example  of  this  kind  is  furnished  by  problem 
No.  23,  p.  71  : 

To  prove  that  the  medians  of  a  triangle  miect  in  one  point. 

If  a,  h,  c  represent  the  three  sides  of  the  triangle,  and  we 
take  as  axes  the  sides  a  and  b,  then  the  equations  of  the  sides 
and  also  of  the  medians  may  be  written  down  with  great  ease, 
as  follows : 


The  sides, 
The  medians, 

a       b 


y  =  o, 


1=0. 


0, 


X     2y 
a'^  b 


^     y 
a      h 


1. 


X      y 

1  =  U,     -~f 

a      0 


0. 


On  comparing  the  equations  of  the  medians,  we  see  that  if 
we  subtract  the  second  equation  from  the  first,  we  obtain  the 
third ;  therefore  the  three  medians  must  pass  through  the 
same  point  (§  59). 


..-'-"^ 


Fig.   39. 


Polar  Co-ordinates. 

91.  There  is  another  system  of  co-ordinates,  called  the  Polar 
System,  which  is  often  useful. 

Let  0  (Fig.  39)  be  a  fixed  point,  AOA'  a  fixed  straight 
line,  P  any  point.     Join  OP. 


108 


ANALYTIC    GEOMETRY. 


It  is  evident  that  we  know  the  position  of  P,  provided 
w^e  know  the  distance  OP  and  the  angle  which  OP  forms 
with  OA. 

Thus,  if  we  denote  the  distance  OP  by  p,  and  the  angle 
POA  by  B,  the  position  of  P  is  determined  if  p  and  0  are 
known. 

p  and  0  are  called  the  Polar  Oo-ordinates  of  P ;  0  is  called 
the  Pole  i   OA,  the  Polar  Axis  ;   OP,  the  Kadins  Vector  of  P. 


Fig.  40. 


Every  point  in  a  plane  is  perfectly  determined  by  a  posi- 
tive value  of  p  between  0  and  co,  and  a  positive  value  of  0 
between  0°  and  360°  (or  0  and  27r,  circular  measure).  But 
in  order  to  be  able  to  represent  by  a  single  equation  all  the 
points  of  a  geometric  locus,  it  is  necessary  to  admit  negative 
values  of  p  and  B,  and  to  adopt  conventions  suitable  for  this 
purpose. 

It  is  agreed  that  B  shall  be  considered  positive  when  it  is 
measured  from  the  initial  line,  in  the  opposite  direction  to  that 
of  the  motion  of  the  hands  of  a  watch ;  and  negative  when 
measured  in  the  same  direction  as  this  motion. 

It  is  also  agreed  that  p,  or  OP,  shall  be  considered  positive 
when  it  forms  one  side  of  the  angle  6,  and  negative  when  it 
does  not. 


DIFFERENT    SYSTEMS    OF    CO-ORDINATES. 


109 


For  example,  suppose  that  the  straight  line  POPi  bisects 
the  first  and  third  quadrants,  and  that  in  this  line  we  take 
points  P,  Pi,  at  the  same  distance  OP  —  p  from  0  ;  then 

P  is  the  point  (p,  ^tt)  or  (— p,  ^tt)  or  (— p,  —  Itt), 
Pi  is  the  point  (p,  {tt)  or  (— p,  ^tt)  or  f     p,  —  f  tt),  etc. 


Fig.  41. 


Fig.  42. 


92.    To  find  the  polo/r  equation  of  the  circle. 

(i.)  Let  the  pole  0  be  at  the  centre.  Then,  if  r  denote  the 
radius,  the  polar  equation  is  simply  p^=r. 

(ii.)  Let  the  pole  0  be  on  the  circumference  (Fig.  41),  and 
let  the  diameter  OB  make  an  angle  a  with  the  initial  line  OA. 
Let  P  be  any  point  (r,  0)  of  the  circle.     Join  BP. 

Then  OP  =  OB  cos  BOP, 

or  P  =  2r  cos  (6  — a).  [23] 

If  OB  is  taken  as  the  initial  line,  the  equation  becomes 

P  =  2  r  cos  e.  [24] 

(iii.)  Let  the  po]e  0  be  anywhere,  and  the  centre  the  point 
(/:,  a).     Then  in  the  triangle  OPC'  (Fig.  42) 

OP'  -  2  OP  X  OCX  cos  POC+  OC^  -  CP'  =  0, 

or  p-  —  2  p7c  cos  (0  —  a)  +  k^  —  r'^  =  O,  [25] 

the  "most  general  form  of  the  polar  equation  of  a  circle. 


110  ANALYTIC    GEOMETRY. 

Ex.    25. 

1.  Find  the  distances  from  the  point  P  in  Fig.  38  to  the 
two  axes. 

2.  Prove  that  the  equation  of  a  straight  line,  referred  to 
oblique  axes  in  terms  of  its  intercepts,  is  identical  in  form 
with  [7],  p.  39. 

3.  If  the  straight  line  F.,OP^  (Fig.  39)  bisects  the  second 
and  fourth  quadrants,  what  are  the  polar  co-ordinates  of  the 
points  Pa  and  Pi  ?  Give  more  than  one  set  of  values  in  each 
case. 

4.  Construct  the  following  points  (on  paper,  take  «  =  1  in.)  : 

,.0),    („.^\    f„,-|\    (-a,t.    Ua,-l 


TT     7r\  /  OTT 


a,  -  ,      ::«,  TT,      acos-,  -  ,      a,  —  .      oa. 


o 


(iy    V       /    V       3  3y    V    2 

3a,  ^""Y    Aa,  tan-^^\    Ua.i^xr^^ 

4  .        . 
Note.     The  expression  tan~^  -  in  higher  Mathematics  means  "the 

angle  whose  tangent  is  -•" 

5.  If  pi,  /32  denote  the  two  values  of  p  in  equation  [25], 
p.  109,  prove  that  pi  and  p^  —  ^'^  —  ^'^-  What  theorem  of  Ele- 
mentary Geometry  is  expressed  by  this  equation  (i.)  when 
the  pole  is  outside  the  circle?  (ii.)  when  the  pole  is  inside 
the  circle? 

6.  Through  a  fixed  point  Pin  a  circle  a  chord  ^P  is  drawn, 
and  then  revolved  about  P;  find  the  locus  of  its  middle  point. 

Note.  In  such  problems  as  this  there  is  a  great  advantage  in  using 
polar  equations. 

7.  If  J9  denote  the  distance  from  the  pole  to  a  straight  line, 
a  the  angle  between  'p  and  the  polar  axis,  prove  that  the  polar 
equation  of  the  line  is  p  cos  {B  —  o.)^=  p. 


DIFFEEENT    SYSTEMS    OF    CO-ORDINATES. 


Ill 


Transformation  of  Co-ordinates. 

93.  The  equation  of  the  same  curve  varies  greatly  in  form 
and  simplicity,  according  to  the  system  of  co-ordinates  adopted, 
and  the  position  of  the  fixed  points  and  lines  with  respect  to 
the  curve.  Hence  it  is  sometimes  useful  to  be  able  to  deduce 
from  the  equation  of  a  curve  referred  to  one  system  of  co- 
ordinates its  equation  referred  to  another  system.  This  pro- 
cess is  known  as  the  Transformation  of  Oo-ordinates. 

It  consists  in  expressing  the  old  co-ordinates  in  terms  of 
the  new,  and  then  replacing  in  the  equation  of  the  curve  the 
old  co-ordinates  by  their  values  in  terms  of  the  new  ;  we  thus 
obtain  a  constant  relation  between  the  new  co-ordinates,  which 
will  represent  the  curve  referred  to  the  new  axes. 

94,  To  change  the  origin  to  the  jjoint  {h,  Jc)  without  changing 
the  directio7i  of  the  axes. 


Y 

v 

f 

> 

\m' 

a 

i 

0 

^ 

i 

M 

X 

Fig.   43. 


Let  OX,  0  F  be  the  old  axes,  O'X',  0'  F'  the  new  ;  and 
let  {x,  y),  (.r',  y')  be  the  co-ordinates  of  the  same  point  P, 
referred  to  the  old  and  new  systems  respectively. 

Then  (Fig.  43) 

OA  =  h,  AO'=k,  OM-^x,  MP  =  y,  a ]\V  ■=  x\  3PP  =  y'. 
x=    OA  +AM^OA  +0'jr  =  x'^h. 
y  =  M2r  -f  M'P  =  ^0'  +  J/'P  =  y'  +  ^. 


112 


ANALYTIC    GEOMETRY. 


These  relations  are  equally  true  for  rectangular  and  oblique 
co-ordinates. 

Hence,  to  find  what  the  equation  of  a  curve  becomes  when 
the  origin  is  transferred  to  a  point  (A,  k),  the  new  axes  run- 
ning parallel  to  the  old,  we  must  substitute  for  x  and  y  the 
values  given  above. 

After  the  substitution,  we  may,  of  course,  write  x  and  y 
instead  of  x'  and  y' ;  so  that  practically  the  change  is  effected 
by  simply  writing  x-\-h  in  place  of  x,  y-}-k  in  place  of  y. 

If,  however,  we  wish  to  transform  a  poi7it  (x,  y)  from  the 
old  to  the  new  system,  we  must  write  x  —  h  in  place  of  x  and 
y~k  in  place  of  y. 

95.  To  change  the  equation  of  a  curve  from  one  rectangular 
system  to  another,  the  origin  remaining  the  same. 


Let  {x,  y)  be  a  point  P  referred  to  the  old  axes  OX,  0  Y\ 
(x',  y'),  the  same  point  referred  to  the  new  axes  OX',  0  Y' 
(Fig.  44).     Then 

OM=x,    3IF  =  y,     ON=x\   NP  =  y\ 

Let  the  angle  XOX'=e.  Draw  XQ,  XE  ±  to  PM,  OX, 
respectively  ;  then 

iVTQ  =  Q^^O  =■■  XOR  =  e. 


DIFFERENT    SYSTEMS    OF    CO-ORDINATES.  113 

Hence  OM=OFc-IiM=OE  —  JVQ=OJVco&e~FJVsinO. 

Or  X  =  x'  cos  0  —  y'  sin  6. 

And     PJ/-  J/Q+  QP-  i?A^+  QF=  OjV  sin  0  +  Pi\^cos  6. 

Or  ?/  —  :r'  sin  6  +  y'  cos  ^. 

Therefore,  to  find  what  the  equation  of  a  curve  becomes 
when  referred  to  the  new  axes,  we  must  write 

X  cos  0  —  y  ^mO  for  x,     x  &m6  -{-y  cos  6  for  y. 

96.  To  transform  an  equation  frovi  one  rectangular  system 
to  another,  both  the  origiii  and  the  direction  of  the  axes  being 
changed. 

First  transform  the  equation  to  axes  through  the  new  origin, 
parallel  to  the  old  axes.  Then  turn  these  axes  through  the 
required  angle. 

If  (h,  k)  is  the  new  origin  referred  to  the  old  axes,  0  the 
angle  between  the  old  and  new  axes  of  x,  we  obtain  as  the 
values  of  x  and  y  for  any  point  P,  in  terms  of  the  new  co- 
ordinates, 7    1      f        a         1    ■     n 
'                  x=^  h  +  X  cos  d  —  y  sin  6, 

y  -—k-{-  x'  sin  0  -\-  y'  cos  0. 

In  making  all  these  transformations,  attention  must  be  paid 
to  the  sigris  of  h,  k,  and  0. 

97.  To  transform  an  equation  from  rectangular  to  oblique 
axes,  the  origin  remaining  the  same. 

Let  a,  (i  be  the  angles  formed  by  the  positive  directions  of 
the  new  axes  0X\  0  Y'  (Fig.  45)  with  the  positive  direction  of 
OX.  Let  the  old  co-ordinates  of  a  point  P  be  x,  y  ;  and  tlie 
new  co-ordinates,  x',  y'.  Then  from  the  right  triangles  OFX, 
FQJVwe  readily  obtain  the  formulas 

x  =  x'  cos  a  +  ?/'  cos  /?, 
y  ==  :r' sin  a -f  ?/' sin /?. 

Investigate  the  special  case  when  yg  =  a  +  90°. 


114 


ANALYTIC    GEOMETRY. 


98.    To  change  an  equation  from  'polar  to  rectangular  co- 
ordinates. 


R    X 


M  X 


Fig.  46. 


Let  the  co-ordinates  of  a  point  P  be  x,  7/  referred  to  the 
rectangular  system,  and  p,  0  referred  to  the  polar  system. 

(i.)  Let  the  origin  of  rectangular  co-ordinates  be  the  pole, 
and  let  the  polar  axis  coincide  with  the  axis  of  x. 
Then  (Fig.  46)    03f-^  OF  cos  FOJl 
FM=  OF  sin  FOIL 
Or  X  =  p  cos  6, 

y  =  p  sin  6. 

(ii.)  If  the  pole  is  the  point  (h,  k),  we  have 

X  =  h-^  p  cos  0, 
y  =  k-i-  p  sin  0. 

(iii.)  If  the  pole  coincides  with  the  origin,  but  the  polar 
axis  OA  makes  the  angle  a  with  the  axes  of  x,  we  obtain 

:r  =  p  cos  (^  +  a), 
y  =  psin  (^  +  a). 

(iv.)  If  the  pole   is  the    point   (h,  Jc),   and  the  polar   axis 
makes  the  angle  a  with  the  axis  of  x, 

x  =  h-{-  p  cos  {6  -\-  a), 
y  =  k-\-psm{0-{-  a). 


DIFFERENT    SYSTEMS    OF    CO-ORDINATES.  115 

99.  To  change  an  equation  from  polar  to  rectangular  co- 
ordinates. 

From  the  results  in  cases  (i.)  and  (ii.)  of  §  98  (the  only- 
cases  of  importance),  we  readily  obtain 

In  case  (i.),      i^  =  x^  +  y^  tan  6  ^ ^  -• 

In  case  (ii.),     r'  =  {x  -  hj  +  (y  —  lc)\     tan  ^  --  ^• 

100.  The  degree  of  an  equation  is  not  altered  hy  any  alteration 
of  tJie  axes. 

For,  however  the  axes  may  be  changed,  the  new  equation 

is  always  obtained  by  substituting  for  x  and  y  expressions  of 

the  form  i    7      i  i      r     i   zf     i     r 

ax  -\-  by  -{-  c    and    a'x  -\-  b' y  -\-  c\ 

These  expressions  are  of  the  first  degree,  and  therefore,  if 
they  replace  x  and  ?/  in  the  equation,  the  degree  of  the  equa- 
tion cannot  be  raised.  Neither  can  it  be  lowered ;  for  if  it 
could  be  lowered,  it  might  be  raised  by  returning  to  the 
original  axes,  and  therefore  to  the  original  equation. 

Ex.  26. 

1.  What  does  the  equation  y^  — 4:r-f43/  +  8  =  0  become 
when  the  origin  is  changed  to  the  point  (1,  —  2)  ? 

Transform  the  equation  of  the  circle  (x  —  ay  +  (y  —  ^Y  =  t^^ 
by  changing  the  origin 

2.  To  the  centre  of  the  circle. 

3.  To  the  left-hand  end  of  the  horizontal  diameter. 

4.  To  the  upper  end  of  the  vertical  diameter. 

5.  What  does  the  equation  x^  ■\-y'^^=  r"^  become  if  the  axes 
are  turned  through  the  angle  a  ? 

6.  What  does  the  equation  x^  —  y"^  =  o^  become  if  the  axes 
are  turned  through  —  45°  ? 


116  ANALYTIC    GEOMETRY. 

7.  The  equation  of  a  curve  referred  to  rectangular  axes  is 
X  —  xy  —  y  =-0.  Transform  it  to  a  new  system,  whose  origin 
is  the  point  (—1,  1),  and  whose  axes  bisect  the  angles  formed 
by  the  old  axes. 

8.  Change  the  following  equations  to  polar  co-ordinates, 
taking  the  pole  at  the  origin  and  the  polar  axis  to  coincide 
with  the  axis  of  x  : 

(i.)  x"^  +  2/^  =  cc'^'      (ii-)  -'^^  —  y"^  =, ci^' 

9.  Change  the  equation  x'^=^^ax  to  polar  co-ordinates, 
(i.)  taking  the  pole  at  the  origin  ;  (ii.)  taking  the  pole  at 
the  point  (a,  0). 

10.  Change  the  following  equations  to  rectangular  co-ordi- 
nates, the  origin  coinciding  with  the  pole,  and  the  polar  axis 

.  with  the  axis  of  x  : 

(i.)  p  =  a,     (ii.)  p  =  acosO,     (iii.)  p"^  cos20  =  a^. 

Transform  the  following  equations  by  changing  the  origin 
to  the  point  given  as  a  new  origin  : 

11.  .r  -f  ?/  +  2  =  0  ;  the  new  origin  (—  2,  0). 

12.  2.i^-5y-10  =  0;  the  new  origin  (5,  - 2). 

13.  Sx'  +  4:xy-\-y'—5x-6y-3  =  0;  new  origin  (|, —4). 

14.  x'^  +  y^  —  2x  —  4:y  =  20;  new  origin  (1,  2). 

15.  x'^  —  6xy -{- y^  —  6x -j-2y -{-1=  0  ;   new  origin  (0, —1). 

16.  Transform  the  equation  o:^  — y^-f-  6  =  0  by  turning  the 
axes  through  45°. 

17.  Transform  the  equation  (x -\- y —  2ay  =  4:xy  by  turn- 
ing the  axes  through  45°. 

18.  Transform  the  equation  9 o:^ —  163/^  =144  to  oblique 
axes,  such  that  the  new  axis  of  x  makes  with  the  old  axis 
of  X  a  negative  angle  whose  tangent  =  —  | ;  and  the  new  axis 
of  y  makes  with  the  old  axis  of  x  a  positive  angle  whose  tan- 
gent is  |. 


DIFFERENT    SYSTEMS    OF    CO-ORDINATES.  117 

Ex.  27.      (Revie^w.) 

1.  Find  the  distance  from  the  point  (—2b,  h)  to  the  origin, 
the  axes  making  the  angle  60°. 

2.  The  axes  making  the  angle  m,  find  the  distance  from  the 
point  (1,  —1)  to  the  point  (—1,  1). 

3.  The  axes  making  the  angle  w,  find  the  distance  from  the 
point  (0,  2)  to  the  point  (3,  0). 

Determine  the  distance  between  the  following  points  referred 
to  polar  co-ordinates : 

4.  (a,  6)  and  (5,  0). 

5.  (a,  0)  and  {a,  -  6). 

6.  (a,  6)  and  (-  a,  -  6). 

7.  (2  a,  30°)  and  (a,  60°). 

8.  Show  that  the  polar  co-ordinates  (p,  ^),  {—p,7r-\-6), 
(— p,  O  —  tt)  all  represent  the  same  point. 

9.  Transform  the  equation  8a;'  +  8.ry+4/-f  122'-f  8y+l  =  0 
to  the  new  origin  {—^,  —  i). 

10.  Transform  the  equation  62:"-f-3?/^  — 24a:-f  6=^0  to  the 
new  origin  (2,  0). 

X        11 

11.  Transform  the  equation  -  +  j  =  l  by  changing  the 
origin  to  the  point  (;^'^)  and  turning  the  axes  through  an 
angle  <^,  such  that  tan  <^  = 

12.  Transform  the  equation  17:r' -16:r?/ -f  17?/' =  225  to 
axes  which  bisect  the  axes  of  the  old  system. 

Transform  the  following  rectangular  equations  to  polar 
equations,  the  polar  axis  in  each  case  coinciding  with  the 
axis  of  X,  and  the  pole  being  at  the  point  whose  co-ordinates 
are  given : 


118  ANALYTIC    GEOMETRY. 

13.  ar-\'y'  =  Qax)  the  pole  (0,  0). 

14.  x'-\-y''  =  Qax]  the  pole  (4«,  0). 

15.  y-  —  Gy  -  5.T  +  9  =  0  ;  the  pole  (f ,  3). 

16.  X-  -  ?/-  -  4.1'  —  6y  -  54  =  0  ;  the  pole  (2,  -  3). 

17.  (.1--  +  y'f  =  k\x'  -  f) ;  the  pole  (0,  0). 

Transform  the  following  polar  equations  to  rectangular  axes, 
the  origin  being  at  the  pole  and  the  axis  of  x  coinciding  with 
the  polar  axis  : 

IS.    p''s'm2er=2a\ 

19.  p  =  ksm20. 

20.  p(sin36'  +  cos3^)  =  5^-sln^cos^. 

21.  Through  what  angle  must  the  axes  of  a  rectangular 
system  be  turned  in  order  that  the  new  axis  of  x  may  pass 
through  the  point  (5,  7)  ? 

22.  The  equation  of  a  straight  line  in  rectangular  axes  is 
Ax-\-JBy-{-C=0.  Through  what  angle  must  the  axes  be 
turned  in  order 

(i.)  that  the  term  containing  x  may  disappear  ? 
(ii.)  that  the  term  containing  y  may  disappear  ? 

23.  Deduce  the  following  formulas  for  changing  from  one 
oblique  system  to  another,  the  origin  remaining  the  same  : 

_  x^  sin  (oi>  —  g)    ,   y'  sin  (oj  —  /S) 
sin  w  sin  w 

x'  sin  a  ,   ?/'  sin  B 
sin  w  sm  (tj 

Note.  In  these  formulas  w  denotes  the  angle  formed  by  the  old  axes, 
a  and  )8  those  formed  by  the  positive  directions  of  the  new  axes  with  the 
positive  direction  of  the  old  axis  of  x. 

24.  From  the  formulas  of  No.  23  deduce  those  of  §  97. 


CHAPTER  y. 

THE    PARABOLA. 

The  Equation  of  the  Parabola. 

101.  A  Parabola  is  the  locus  of  a  point  whose  distance  from 
a  fixed  point  is  always  equal  to  its  distance  from  a  fixed 
straight  line. 

The  fixed  point  is  called  the  Focus ;  the  fixed  straight  line, 
the  Directrix. 

The  straight  line  which  passes  through  the  focus,  and  is  per- 
pendicular to  the  directrix,  is  called  the  Axis  of  the  parabola. 

The  intersection  of  the  axis  and  the  directrix  is  called  the 
Foot  of  the  axis. 

The  point  in  the  axis  half  way  betw^een  the  focus  and  the 
directrix  is,  from  the  definition,  a  point  of  the  curve  ;  this 
point  is  called  the  Vertex  of  the  parabola. 

The  straight  line  joining  any  point  of  the  curve  to  the  focus 
is  called  the  Focal  Kadius  of  the  point. 

A  straight  line  passing  through  the  focus  and  limited  by 
the  curve  is  called  a  Focal  Chord. 

The  focal  chord  perpendicular  to  the  axis  is  called  the  Latus 
Seotum  or  Parameter. 

102.  To  construct  a  parabola,  having  given  the  focus  and 
the  directrix, 

I.    Bij  Points.     Let  i^(Fig.  47)  be  the  focus,  C^the  direc-  . 
trix.     Draw  the  axis  FB,  and  bisect  FB  in  A  ;  then  A  is  the 
vertex  of  the  curve.     At  any  point  3f  in  the  axis  erect  a  per- 
pendicular.    From  F  as  centre,  with  DM  as  radius,  cut  this 


-^ 


120 


ANALYTIC    GEOMETEY. 


perpendicular  in  P  and  Q ;  then  P  and  Q  are  two  points*  of 
the  curve,  for  FP=  DM=  distance  of  P  or  Q  from  CE.  In 
the  same  way  we  can  find  as  many  points  of  the  curve  as  we 
please.  After  a  sufficient  number  of  points  has  been  found, 
we  draw  a  smooth  curve  through  them. 


Fig.  47. 


Fig.  48. 


II.  By  Motion.  Place  a  ruler  so  that  one  of  its  edges 
shall  coincide  with  the  directi-ix  DE  (Fig.  48).  Then  place 
a  triangular  ruler  BCE  with  the  edge  CE  against  the  edge 
of  the  first  ruler.  Take  a  string  whose  length  is  equal  to 
BC\  fasten  one  end  at  B  and  the  other  end  at  F.  Then 
slide  the  ruler  BCE  along  the  directrix,  keeping  the  string 
tightly  pressed  against  the  ruler  by  the  point  of  a  pencil  P. 
The  point  P  will  trace  a  parabola  ;  for  during  the  motion  we 
always  have  PF=  PC 

103.  To  find  the  equation  of  the  'parctbola,  when  its  axis  is 
taken  as  the  axis  of  x  and  its  vertex  as  the  origin. 

Let  F  (Fig.  49)  be  the  focus,  CE  the  directrix,  DFX  the 
axis,  A  the  vertex  and  origin ;  also  let  2f  denote  the  known 
distance  FD. 

Let  P  be  any  point  of  the  curve  ;  then  its  co-ordinates  are 

AM=x,     P2I=y. 


THE    TARABOLA. 


121 


•Draw  FC 1.  to  CE  \  then  by  the  definition  of  the  curve 
FF  =FC=-DM. 


Therefore 

Now 
and 


Fr^=DM\ 

FF'  =  Fir  +  FM'  =  y'+ix-py 


Whence  y'  =  4^x.      y-   ^^  ^  [26] 

This  is  called  the  principal  equation  of  a  parabola. 


Therefore       3/^  +  (:i'  —  p)y  =  (x  -j-pf. 


104.  Since  y^  and  p  in  equation  [26]  are  positive,  x  must 
always  be  positive  ;  therefore  the  curve  lies  wholly  on  the 
positive  side  of  the  axis  of  y. 

A  further  examination  of  equation  [26]  shows  that  the 
curve,  (i.)  passes  through  the  origin,  (ii.)  is  symmetrical  with 
respect  to  the  axis  of  .r,  (iii.)  extends  towards  the  right  with- 
out limit,  and  (iv.)  recedes  from  the  axis  of  x  without  limit. 

105.  Any  point  (h,  k)  is  outside,  on,  or  inside  the  parabola 
y^  ■=  4ipx,  according  as  k'^  —  4:ph  is  positive,  zero,  or  negative. 

Let  Q  be  the  point  (A,  ^),  and  let  its  ordinate  meet  the 
curve  in  F. 


122  ANALYTIC    GEOMETRY. 

If  Ic^  —  4:ph  =  0,  the  point  (A,  k)  satisfies  equation  [2G],  and 
therefore  Q  coincides  with  P. 

If  ¥  —  ^'ph  is  positive,  or  Ji^  >  4^A,  then,  since  P3I  ^=4:ph, 
we  have  QM' >  PM'\  or  QiV>FM;  hence  Q  is  outside 
the  curve. 

If  k'^  —  ^ph  is  negative,  we  may  prove  similarly  that  Q 
must  be  inside  the  curve. 

106.  To  find  the  latus  rectum  of  a  parabola. 

The  common  abscissa  of  the  two  points  where  the  latus 
rectum  meets  the  curve  =^.  Substituting  this  value  for  x 
in  equation  [26],  we  have  y  =  ±2p.  Therefore  the  latus 
rectum  =  ^p. 

107.  To  find  the  points  in  which  the  straight  line  i/  =  mx-\-c 
meets  the  parabola  y^  =  ^px. 

The  co-ordinates  of  these  points  must  satisfy  both  equations  ; 
hence,  at  a  common  point,  we  have  the  relation 

\     771     j 

Since  (1)  is  a  quadratic  equation,  we  see  that  every  straight 
line  meets  a  parabola  in  two  points.  Solving,  (1).  we  obtain 
for  the  ordinates  of  these  two  points 


•^  7?l  m     \  rp 


l^LZ^;  (2) 


whence   it  appears  that   the   points   are  real,   coincident,   or 
imaginary,  accordmg  2C$,p  —  mc  is  positive,  zero,  or  negative. 

108.  To  find  the  equation  of  a  parabola  ivhose  axis  is  par- 
allel to  the  axis  of  x. 

Let  the  vertex  be  the  point  (a,  5),  and  let  2p  —  distance 
from  focus  to  directrix.  Then  the  focus  will  be  the  point 
{a-\-p,  b),  and  the  directrix  will  be  the  line  x  =  a  —p. 


THE    FARABOLA.  123 

The  distance  of  any  point  {x,  y)  from  the  focus  is 


and  its  distance  from  the  directrix  is 

X  —  Ci^lp. 

If  {x,  y)  is  a  point  of  the  parabola,  these  distances  are  equal ; 
putting  them  equal,  and  reducing,  we  obtain 

y2  _  4^^  _  O^y  _|_  ^2  _  4^^  ^  Q  (2) 

Hence  we  may  infer  that  in  general  an  equation  having 
theform      ,f  j^  Ax -\- By -^  C  =^^  \^^r  (2) 

represents  a  parabola  having  its  axis  parallel  to  the  axis  of  x. 
By  equating  coefficients  in  (1)  and  (2),  we  obtain 

.  .  B'-^C     -,  B 


whence  the  following  results  easily  follow  : 
The  latus  rectum  =  —. 
The  vertex  is  the  point 


The  latus  rectum  =^  —A. 

fB'~4:C    _B 

[      4:A      '  2 

ine  locus  is  the  point . • 

p 

The  axis  is  the  line  y  = 

2 

ine  directrix  is  the  line  x  = ■ 

4:A 

If  A  is  negative,  the  parabola  lies  to  the  right  of  the  axis 
of  ?/,  and  may  be  called  right-ho,nded. 

If  A  is  positive,  the  parabola  lies  to  the  left  of  the  axis  of  ?/, 
and  may  be  called  left-handed. 


124  ANALYTIC    GEOMETRY. 


Ex.  28. 


1.  Show  that  the  distance  of  any  point  of  the  parabola 
y^  =  4:2^x  from  the  focus  is  ec|ual  to  ^)  +  x. 

2.  Find  the  equation  of  a  parabola,  taking  as  axes  the 
axis  of  the  curve  and  the  directrix. 

3.  Find  the  equation  of  a  parabola,  taking  the  axis  of  the 
curve  as  the  axis  of  x  and  the  focus  as  the  origin. 

4.  The  distance  from  the  focus  of  a  parabola  to  the  direc- 
trix =  5.     "Write  its  equation, 

(i.)  If  the  origin  is  taken  at  the  vertex, 
(ii.)  If  the  origin  is  taken  at  the  focus, 
(iii.)  If  the  axis  and  directrix  are  taken  as  axes. 

5.  The  distance  from  the  focus  to  the  vertex  of  a  parabola 
is  4.  Write  its  equations  for  the  three  cases  enumerated  in 
No.  4. 

6.  For  what  point  of  the  parabola  y^^^lSx  is  the  ordi- 
nate equal  to  three  times  the  abscissa  ? 

7.  Find  the  latus  rectum  for  the  following  parabolas  : 

Find  the  points  common  to  the  following  parabolas  and 
straight  lines  : 

y-  =  ^x,     3.r  — 7y  +  30  =  0. 


?/---3.r,     :r- 43/ +  12  =  0. 

?/^  =  4:r,     .'r  =  9,     2:  =  0,     a;  =  —  2. 

?/  =  4.i',     y  =  6,     y  =  —  8. 

What  must  be  the  value  of  ^j  in  order  that  the  parabola 


?/  =  ^po:  may  pass  through  the  point  (9,  —12)  ? 


THE    PARABOLA.  125 

13.  For  what  point  of  the  parabola  y-  =  2>2,x  is  the  ordi- 
nate equal  to  4  times  the  abscissa  ? 

•  14.  The  equation  of  a  parabola  is  y'  =  Qx.  What  is  the 
equation  of  (i.)  its  axis,  (ii.)  its  directrix,  (iii.)  its  latus  rectum, 
(iv.)  a  focal  chord  through  the  point  whose  abscissa  =  8,  (v.)  a 
chord  passing  through  the  vertex  and  the  negative  end  of  the 
latus  rectum  ? 

15.  The  equation  of  a  parabola  is  y^  =  16a;.  Find  the 
equation  of  (i.)  a  chord  through,  the  points  whose  abscissas 
are  4  and  9,  and  ordinates  positive ;  (ii.)  the  circle  passing 
through  the  vertex  and  the  ends  of  the  latus  rectum. 

IG.  If  the  distance  of  a  point  from  the  focus  of  the  parabola 
y2  —  4^;-^.  is  equal  to  the  latus  rectum,  what  is  the  abscissa  of 
the  point  ? 

17.  In  the  parabola  if^^^jpx  an  equilateral  triangle  is 
inscribed  so  that  one  vertex  is  at  the  origin.  What  is  the 
length  of  one  of  its  sides? 

18.  A  double  ordinate  of  a  parabola  =  8p.  Prove  that 
straight  lines  drawn  from  its  ends  to  the  vertex  are  perpen- 
dicular to  each  other. 

Explain  how  to  construct  a  parabola,  having  given 

19.  The  directrix  and  the  vertex. 

20.  The  focus  and  the  vertex. 

21.  The  axis,  vertex,  and  latus  rectum. 

22.  The  axis,  vertex,  and  a  point  of  the  curve. 

23.  The  axis,  focus,  and  latus  rectum. 

24.  The  axis,  directrix,  and  one  point. 

25.  The  axis  and  two  points. 


126  ANALYTIC   GEOMETRY. 

26.  Determine,  as  regards  size  and  position,  the  relations 
of  the  following  parabolas  : 

(^i.)  7/^=z4:px,  (ii.)  y^---=^—4:px,  (iii.)  a;^^4^y,  (iv.)  :r^  =  — 4^5?/. 

27.  What  is  the  locus  of  the  equation  y'^  -{-  Ax  ^  B// -{-  C  —  0 
in  the  following  special  cases : 

(i.)    A  =  0?        (in.)C=0?  (v.)    A  =  C=0? 

(ii.)  ^  =  0?       (iv.)^=^  =  0?      (yI)  B  =  C=0? 

28.  Show  that  in  general  the  equation  x^-\-Ax-\-  By-^  C=  0 
represents  a  parabola  whose  axis  is  parallel  to  the  axis  of  y ; 
and  determine  the  latus  rectum,  the  vertex,  the  focus,  the 
axis,  and  the  directrix. 

Find  the  latus  rectum,  vertex,  focus,  axis,  and  directrix  of 
the  following  parabolas  : 

29.  ?/--125;-f  84-=0.  .      33.  :c' -  12y  +  84  =  0. 

30.  /-12^'-84-=0.  34.  ?/-- 8^^  -  8?/  + 64  =  0. 

31.  y^  +  12:^  +  84  =  0.  35.  l+22'  +  3y-  =  0. 

32.  2/2 +  12:^-84  =  0.  36.  y  =  x'~x-2. 

37.   ?/'-4:r  +  6y  +  l  =  0. 

Tangents  and  Normals. 

109.  To  find  the  equation  and  the  normal  of  the  tangent  to 
the  rparahola  2/^^=  ^px,  at  the  -point  of  contact  {x^,  y^). 

If  P,  Q  are  the  points  (.Tj,  ?/i),  (o;^,  y.^,  the  equation  of  a 
straight  line  through  P  and  Q  is 


If  P  and  Q  are  points  of  the  parabola  y^  =  4:px, 
y^  =  4:pxi, 
2/2'  =  ^pXi. 


(1) 


127 


THE    PARABOLA. 

Whence  ^^iZJ^^-=  _i£_. 

^2-^'i     3/2 +  yi 

By  substitution,  equation  (1)  becomes 

x-x^      y,  +  yi ' 

whence,    by    clearing    of    fractions,    and    remembering    that 
yx  =  ^px,  we  obtain  the  equation  of  a  secant  in  the  form 


y(yi  +  y-i)  —  yiy-i  =  ^px. 


(2) 


Y 

c 

'■^'.^ 

.^ 

\ 

.^^ 

\ 

^ 

\ 

E,.^-^'^y^ 

7 

\ 

\ 

-^"^                        \ 

D 

A 

U' 

M 

N 

X 

•  \ 

'    • 

Fig.  50. 

Now  let  the  secant  turn  about  F  till  Q  coincides  with  F ; 
then  x\  =  Xi,  y2  =  yi,  the  secant  becomes  the  tangent  at  P, 
and  its  equation  reduces  to 

ViV  =  2p{3c  +  X,).  [27] 

The  normal  passes  through  (x^,  y^),  and  is  perpendicular  to 
the  tangent ;  hence  (§  50)  its  equation  is 


'PiU  —  Vi)  -hvA^  —  oc, 


O. 


[28] 


128  ANALYTIC    GEOMETEY. 

110.  If  we  make  y  =  0  in  equations  [27]  and  [28],  we  obtain 

x^  —  x^   and   x=^Xi-\- 2p. 
Hence  the  values  of  the  subtangent  MT  and  the  subnormal 

Therefore 

(i.)    The  subtangent  is  bisected  at  the  vertex. 
(ii.)  The  subnormal  is  constant,  and  equal  to  the  distance 
from  the  focus  to  the  directrix. 

111.  In  the  triangle  FFT  (Fifr,  50)  we  have 

FT=  FA  +  A  T=p  +  X, 

FF=  FC  =  FM=FA  +AM=j)  +  x. 

Therefore         FT=  FF. 

Hence  the  angle 

FFT=^  FTF=  TFa  or 

The  tangent  to  a  parabola  at  any  point  makes  equal  angles 
with  the  focal  radius  and  the  line  passing  through  the  point 
parallel  to  the  axis. 

112.  To  find  the  equation  of  a  tangent  to  a  parabola  in  ten-tns 
of  its  slope. 

From  the  result  obtained  in  §  107  we  see  that  the  straight 
line  y  =  mx  +  c  touches  the  parabola  y"^  =  4^9^,  when 

mc^=p, 

P 
or  c  =  —- 

tn 

Hence,  for  all  values  of  m,  the  straight  line 


11  =  inx  +  — 
^  m 


will  touch  the  parabola  y"  =  ^p)x. 


THE    PARABOLA.  129 

Ex.  29. 

1.  The  normal  to  a  parabola  at  any  point  bisects  the  angle 
between  the  focal  radius  and  the  line  drawn  through  the 
point  parallel  to  the  axis. 

Note.  The  use  of  parabolic  reflectors  depends  on  this  property.  A 
ray  of  light  issuing  from  the  focus  and  falling  on  the  reflector  is  reflected 
in  a  line  parallel  to  the  axis  of  the  reflector. 

2.  Explain  how  to  draw  a  tangent  and  a  normal  to  a  given 
parabola  at  a  given  point. 

3.  Prove  that  FC  (Fig.  50)  is  perpendicular  to  PT. 

4.  Prove  that  the  tangent  y  =  mx  +  —  touches  the  parabola 


y-  =  ^px  at  the  point  f  — i,'  — 1* 


6.  Prove  that  the  equation  of  a  normal  to  the  parabola 
y"^  =  4lPX  in  terms  of  its  slope  is  y  =  ^nix  —  '???^;(2  -{-  vi^). 

6.  What  are  the  equations  of  a  tangent  and  a  normal  to 
the  parabola  y^  =  5.r,  passing  through  the  jDoint  whose  abscissa 
is  20  and  ordinate  positive  ? 

7.  AVhat  are  the  equations  of  the  tangents  and  the  normals 
to  the  parabola  y'  =  12x,  drawn  through  the  ends  of  the  latus 
rectum  ?     Find  the  area  of  the  figure  which  they  enclose. 

8.  Given  the  parabola  y'  =  \Ox.  Through  the  point  whose 
abscissa  is  7  and  ordinate  positive  a  tangent  and  a  normal  are 
drawn.  Find  the  lengths  of  the  tangent,  the  normal,  the 
subtangent,  and  the  subnormal. 

9.  A  tangent  to  the  parabola  ?/^  =  20:r  makes  Avith  the  axis 
of  x  an  angle  of  45°.     Determine  the  point  of  contact. 

10.  Show  that  the  focus  i^(Fig.  50)  is  equidistant  from  the 
points  P,  T,  N.  What  easy  way  of  drawing  a  tangent  and  a 
normal  is  suggested  by  this  theorem  ? 


130  ANALYTIC    GEOMETRY. 

11.  If  i^is  the  focus  of  a  parabola,  and  Q,  H  denote  the 
points  in  which  a  tangent  cuts  the  directrix  and  the  latus 
rectum  produced,  prove  that  FQ  =  FB. 

12.  Prove  that  tangents  drawn  through  the  ends  of  the 
latus  rectum  are  J_  to  each  other. 

13.  Find  the  distances  of  the  vertex  and  the  focus  from  the 

P 
tangent  y  =  mx  -f-  — • 

14.  The  points  of  contact  of  two  tangents  are  {x^,  y^  and 
(x2, 3/2)-     Find  their  point  of  intersection. 

15.  A  tangent  to  the  parabola  3/^  =  4:px  cuts  equal  inter- 
cepts on  the  axes.  What  is  its  equation  ?  What  is  the  point 
of  contact  ?     What  is  the  value  of  the  intercept  ? 

16.  Through  what  point  in  the  axis  of  x  must  tangents  to 
the  parabola  y^  =  4:px  be  drawn  in  order  that  they  may  form 
with  the  tangent,  through  the  vertex,  an  equilateral  triangle  ? 

17.  For  what  point  of  the  parabola  y'^  =  ^px  is  the  normal 
equal  to  twice  the  subtangent? 

18.  For  what  point  of  the  parabola  y'^=^^px  is  the  normal 
equal  to  the  difference  between  the  subtangent  and  the  sub- 
normal ? 

19.  Find  the  equation  of  a  tangent  to  the  parabola  ?/  =  5:r 
parallel  to  the  straight  line  3  a;  —  2?/ -1-7=  0.  Also  find  the 
point  of  contact. 

20.  Find  the  equation  of  the  straight  line  which  touches 
the  parabola  y'  =  Vlx  and  makes  an  angle  of  45°  with  the 
line  y  =  3a;  —  4.     Also  find  the  point  of  contact. 

21.  Find  the  equation  of  a  straight  line  which  touches  the 
parabola  y^^lGo;  and  passes  through  the  point  (~4,  8). 

22.  If  a  normal  to  a  parabola  meet  the  curve  again  in  the 
point  Q,  find  the  length  of  PQ. 


THE    PARABOLA.  131 

23.  Prove  by  the  secant  method  that  the  equation  of  a  tan- 
gent to  the  parabola  'if'  =  4:px  —  ^p"^,  at  the  point  (xi,  y^  is 

24.  Find  the  equations  of  the  tangents  and  normals  to  the 
parabola  y^  —  8:?;  —  6?/ —  63  =  0,  drawn  through  the  point.s 
whose  common  abscissa  =  —  1. 

25.  What  are  the  equations  of  tangents  to  the  following 
parabolas  : 

(i.)  7/^-  4:2?x  ?       (ii.)  X-  =  4:2^1/  ?       (iii.)  x'  =  -  4j9y  ? 


Ex.  30.      (He^iew.) 

Note.     If  not  otherwise  specified,  the  axis  of  the  parabola  and  the 
tangent  at  the  vertex  are  to  be  assumed  as  axes  of  co-ordinates. 

What  is  the  equation  of  a  parabola, 

1.  If  the  axis  and  directrix  are  taken  as  axes,  and  the  focus 
is  the  point  (12,  0)  ? 

2.  If  the  axis  and  tangent  at  the  vertex  are  the  two  axes, 
and  (25,  20)  is  a  point  on  the  curve  ? 

3.  If  the  same  axes  are  taken,  and  the  focus  is  the  point 
(-4i,0)? 

4.  If  the  axis  is  taken  as  the  axis  of  x,  the  vertex  is  the 
point  (5,  —  3),  and  the  latus  rectum  =  5|-? 

5.  If  the  axis  is  the  line  y  =  —  7,  the  abscissa  of  the  vertex 
=  3,  and  one  point  is  (4,  —  5)  ? 

6.  If  the   curve  passes    through    the   points  (0,  0),  (3,  2), 
(3,  -  2)  ? 

7.  If  the    curve   passes  through  the  points  (0,  0),  (3,  2), 
(-3,2)? 


132  ANALYTIC    GEOMETRY. 

8.  What  is  the  latus  rectum  of  the  parabola  2?/^  =  3 a:? 
What  is  the  equation  of  its  directrix,  and  of  the  focal  chords 
passing  through  the  points  whose  abscissa  =  6  ? 

9.  Describe  the  change  of  form  which  the  parabola  ?/^  =  4pa; 
undergoes  as  we  suppose  ^:>  to  diminish  without  limit. 

10.  Find  the  intercepts  of  the  parabola  7/^+4.-^;  — 6?/— 16  =  0. 

11.  One  vertex  of  an  equilateral  triangle  coincides  with  the 
focus,  and  the  others  lie  in  the  parabola  if  =  A:]px.  Find  the 
length  of  one  side. 

12.  The  latus  rectum  of  a  parabola  =  8  ;  find 

(i.)  Equation  of  a  tangent  through  its  positive  end. 
(ii.)  Distance  from  the  focus  to  this  tangent, 
(iii.)  Equation  of  the  normal  at  this  point. 

13.  What  is  the  equation  of  the  chord  passing  through  the 
two  points  of  the  parabola  y'^=^x  for  which  Xi  =  2,  3/1  >  0, 
and  :i'2  =  18,  y2<0? 

14.  Find  the  equation  of  the  chord  of  the  parabola  'f^^'ipx 
which  is  bisected  at  a  given  point  (:i'i,  ?/i). 

15.  In  what  points  does  the  line  x-\-y  —  12  meet  the  para- 
bola 2/^  +  2:r-12y  +  16  =  0? 

16.  In  what  points  does  the  line  ^y='2x-\-d>  meet  the 
parabola  ?/-  — 4.'u  —  8?/  + 24  =  0? 

17.  Find  the  equations  of  tangents  from  the  origin  to  the 
parabola  {y  —  by  =  4ip(x  —  a). 

18.  Describe  the  position  of  the  parabola  y^  +  2.r  +  4  =  0 
with  respect  to  the  axes,  and  determine  its  latus  rectum, 
vertex,  focus,  and  directrix. 

19.  What  is  the  distance  from  the  origin  to  a  normal  drawn 
through  the  end  of  the  latus  rectum  of  the  parabola 

y^  =  4  a(x  —  a)? 


THE    PARABOLA.  133 

Find  the  equation  of  a  parabola, 

20.  If  the  equation  of  a  tangent  is  4y  =  3:r  —  12. 

21.  If  a  focal  radius  =  10,  and  its  equation  is  3?/  =  4:r  — 8. 

22.  If  for  a  point  of  the  curve  the  focal  radius  =  r,  the 
length  of  the  tangent  =  t. 

23.  If  for  a  j^oint  of  the  curve  the  focal  radius  ■=  r,  the 
length  of  the  normal  =  n. 

24.  If  for  a  point  of  the  curve  the  length  of  the  tangent  =  i, 
the  length  of  the  normal  =  n. 

25.  If  for  a  point  of  the  curve  the  focal  radius  =  r,  the 
subtangent  =  s. 

26.  Two  parabolas  have  the  same  vertex,  and  the  same 
latus  rectum  4^,  but  their  axes  are  _L  to  each  other.  What 
is  the  length  of  their  common  chord  ? 

27.  Through  the  three  points  of  the  parabola  3/'  =  12:r, 
whose  ordinates  are  2,  3,  6,  tangents  are  drawn.  Show  that 
the  circle  circumscribed  about  the  triangle  formed  by  the 
tangents  passes  through  the  focus. 

28.  A  tangent  to  the  parabola  y^  =  "ipx  makes  the  angle 
30°  with  the  axis  of  x.     At  what  point  does  it  cut  the  axis  ? 

29.  For  what  point  of  the  parabola  y"^  =  4j9:r  is  the  length 
of  the  tangent  equal  to  4  times  the  abscis.sa  of  the  point  of 
contact? 

30.  The  product  of  the  tangent  and  normal  is  equal  to 
twice  the  square  of  the  ordinate  of  the  point  of  contact.  Find 
the  point  of  contact  and  the  inclination  of  the  ordinate  to  the 
axis  of  X. 

31.  Two  tangents  to  a  parabola  are  perpendicular  to  each 
other.     Find  the  product  of  their  subtangents. 


134  ANALYTIC    GEOMETRY. 

32.  Prove  that  the  circle  described  on  a  focal  radius  as 
diameter  touches  the  tangent  drawn  through  the  vertex. 

33.  Prove  that  the  circle  described  on  a  focal  chord  as 
diameter  touches  the  directrix. 

Find  the  locus  of  the  middle  points 

34.  Of  all  the  ordinates  of  a  parabola. 

35.  Of  all  the  focal  radii. 

36.  Of  all  the  focal  chords. 

37.  Of  all  chords  passing  through  the  vertex. 

38.  Of  all  chords  that  meet  at  the  foot  of  the  axis. 

Two  tangents  to  the  parabola  y^  =  ^'px  make  the  angles 
B,  6^  with  the  axis  of  x  ;  find  the  locus  of  their  intersection 

39.  If  cot  ^  +  cot  ^' -  Z;.  41.    If  tan  ^  tan  ^' =  Z:. 

40.  If  cot  6  -  cot  &  =  h  42.    If  sin  0  sin  0^  =  h 

43.  Find  the  locus  of  the  centre  of  a  circle  which  passes 
through  a  given  point  and  touches  a  given  straight  line. 

SUPPLEMENTARY   PROPOSITIONS. 

113.  Two  tangents  can  he  draivn  to  a  j'JCirabola  from  any 
point ;  and  they  ivill  he  real,  coincident,  or  imaginary,  accord- 
ing as  the  point  is  luithout,  on,  or  within  the  curve. 

The  tangent  y  =  mx  +  —  will  pass  through  the  point  (h,  J:) 
if  ""' 

Z-  =  mh  +  ^  ; 

that  is,  if  hni"  -Jcm+p^O; 

whence  ^,,  ^  >^  ±  V^"'^  -  4_^. 

2^; 


THE    PARABOLA.  135 

Since  there  are  two  values  of  m,  two  tangents  can  be  drawn 
through  the  point  (A,  k). 

The  values  of  m  are  real,  coincident,  or  imaginary,  accord- 
ing as  k"^  —  4  hp  is  positive,  zero,  or  negative  ;  that  is  (§  105), 
according  as  (h,  k)  is  without,  on,  or  within  the  curve. 

114.  To  find  the  equation  of  the  straight  line  through  the 
points  of  contact  of  the  two  tangents  drawn  to  the  parabola 
y'  =  ^px  from  the  point  (h,  k). 

If  (xi,  ?/i)  and  (x-i,  y.J  are  the  points  of  contact,  the  equa- 
tions of  the  tangents  are 

y,y  =  2p(x  +  Xi), 
y.{y  =  2p{x  +  x.^. 

Since  (A,  k)  is  in  both  these  lines, 

hj,=^2p{x,-\-h),  (1) 

kij,  =  2p(x,  +  A).  (2) 

But  equations  (1)  and  (2)  are  the  conditions  which  make 
the  points  (xi,  y^)  and  (0:2,  y.^)  lie  in  the  straight  line  whose 
equation  is 

Hence  (3)  is  the  equation  recjuired. 

115.  The  straight  line  joining  the  points  of  contact  of  the 
two  tangents  (real  or  imaginary)  from  any  point  P  to  a  para- 
bola is  called  the  Polar  of  P  with  respect  to  the  ^^arabola  ;  and 
the  point  P  is  called  the  Pole  of  the  straight  line  with  respect 
to  the  parabola. 

The  propositions  in  §§  80-82,  relating  to  poles  and  polars 

with  respect  to  a  circle,  also  hold   true  for  poles  and  polars 

with  respect  to  a  parabola,  and  may  be  proved  in  exactly  the 
same  way. 


136 


ANALYTIC    GEOMETRY. 


116.    To  find  tlic  locus  of  the  middle  points  of  parallel  chords 
in  the  parabola  y*  =  4ipx. 


Fig.  51. 


Let  the  equation  of  any  one  of  the  chords  FQ  (Fig.  51)  be 
y  =  mx  +  c,  and  let  it  meet  the  curve  in  the  points  (xi,  ?/i), 

-- ^-  (1) 


Then  (§  109) 


m 


2/1  +  2/2 


Let  (x,y)  be  its  middle  point  M;  then  2^  =  yi -{- y.^.     By 
substitution  in  (1)  we  obtain 

2p 


y 


(2) 


a  relation  which  holds  true  for  all  the  chords,  because  m  is 
the  same  for  all  the  chords.  The  required  locus,  therefore, 
is  represented  by  (2),  and  is  a  straight  line  parallel  to  the 
axis  of  X. 

The  locus  of  the  middle  points  of  a  system  of  parallel  chords 
in  a  parabola  is  called  a  Diameter  ;  and  the  chords  are  called 
the  Ordinates  of  the  diameter. 


THE    TARABOLA.  137 

Therefore  every  diameter  of  a  parabola  is  a  straight  line 
parallel  to  its  axis. 

Conversely,   every  straight  line  parallel  to  the  axis  is  a 

diameter  ;  for  ??z,  and  therefore  ■ — >  may  have  any  value  what- 
ever. 

117.  Let  the  diameter  through  M  meet  the  curve  at  8,  and 
conceive  the  straight  line  PQ  to  move  parallel  to  itself  till  P 
and  Q  coincide  at  S\  then  the  straight  line  becomes  the  tan- 
gent at  8 ;  therefore 

The  tangent  drawn  through  the  extremfiity  of  a  diameter  is 
parallel  to  the  ordinates  of  the  diameter. 

118.  From  the  focus  F  draw  FC  ±  to  PQ,  and  let  FC 
meet  the  directrix  in  the  point  C.  If  6  denote  the  angle 
which  the  chord  PQ  makes  with  the  axis  of  x,  it  easily  follows 
that  DCF=  0  ;  then  we  have 

(77)  =  i^D  c o t  (9  ^  —  ^  ^'  o    •' ;  that  is, 
m  2       ' 

The  perpendicular  to  a  chord  which  passes  through  the  focus 
meets  the  diameter  of  the  chord  in  the  directrix. 

119.  Let  the  tangents  drawn  through  P  and  Q  meet  in  the 
point  T.     By  solving  their  equations, 

y,y  =  2p{x  +  a;,), 
y.,y=r.2p{x-\-x.;), 

we  obtain  for  the  value  of  the  ordinate  of  T 

2p(x^  —  x\)      2p      3/i  +  ?/2      ^^ 
y  — =  —  =  — o — .     Hence 

Tangents  drawn  through  the  ends  of  a  chord  meet  in  the 
diameter  of  the  chord. 


138  ANALYTIC    GEOMETRY. 

120.  What  is  the  locus  of  the  foot  of  a  perpendicular  dropped 
from  the  focus  to  a  tangent  ? 

Let  the  equation  of  tlie  tangent  be 
y  =  mx  -4-  ^^' 

Then  the  equation  of  the  perpendicular  will  be 

X       p 

y  = h-- 

^  m      m 

Since  these  two  lines  have  the  same  intercept  on  the  axis 
of  y,  they  meet  in  that  axis  ;  that  is,  in  the  tangent  through 
the  vertex.     This  tangent,  therefore,  is  the  required  locus. 


121.    The  perpendicular 


X      p 
^  rn,      m 


meets  the  directrix  in  the  point  i—p,  ^-).     But  the  ordinate 


T) 

of  the  point  where  the  tangent  y  =  mx  +  —  meets  the  para- 
bola  y'^  =  A:px  is  also  — ;  therefore 

The  perpendicular  from  the  focus  to  a  tangent,  if  produced, 
meets  the  directrix  in  the  diameter  through  the  point  of  contact. 

122.    The  distance  of  anv  point  (A,  1:)  from  the  focus  {p,  0)  i^ 


and  its  distance  from  the  point  (  —  7^  —  )  is 


>J(^+^)'+(^-? 


THE    PARABOLA.  139 

If  (A,  7")  is  in  the  tangent  y  =  mx  +  — » 

then  k  —  mh  -f  —• 

Now  this  value  of  h,  substituted  in  the  two  expressions  for 
distances  just  given,  makes  them  equal ;  therefore 

Every  point  in  the  tangent  is  equidistant  from  the  focus  and 
the  2^oint  where  the  perpendicular  from  the  focus  to  the  tangent 
viects  the  directrix. 

123.  What  is  the  locus  of  the  intersection  of  two  perpendicular 
tangents  f 

If  the  equation  of  one  tangent  is 

p 

y  =:^  mx  A » 

^  '   ni 

then  the  equation  of  the  other  is  found  to  be 
Subtracting  one  equation  from  the  other, 


(a;+i5)(^m+-j  =  0. 


But  m  -j is  not  0  ;  therefore 

m 

x+p  =  0, 

or  x=^—p, 

the  equation  of  the  directrix. 

Hence  the  directrix  is  the  locus  required. 

124.    Tangents  are  drawn  through  the  ends  of  a  focal  chord. 
What  is  the  locus  of  their  intersection  ? 

Let  (/i,  h)  be  their  intersection  ;    then  we  may  write  the 
equation  of  the  chord     ^^  _  ^^^^  ^  ^^^ 


140 


ANALYTIC    GEOMETRY. 


If  the  chord  passes  through  the  focus  Q;,  0),  we  have 

whence  h  =  —p. 

Therefore  the  required  locus  is  the  directrix.  And  therefore 
combining  this  result  with  that  obtained  in  §  123,  we  see  that 

Tangents  through  the  ends  of  a  focal  chord  are  perpendicular 
to  each  other. 

125.  To  find  the  equation  of  a  parabola  referred  to  any 
diameter  and  the  tangent  through  its  extremity  as  axes. 

Transform  the  equation  y"^  =  4:px  to  the  diameter  SJ^' 
(Fig.  52)  and  the  tangent  through  jS  as  new  axes.  Let  m  be 
the  slope  of  the  tangent,  0  the  angle  which  the  tangent  makes 
with  the  diameter ;  then  m  =  tan  0. 


Fig.   52. 


First  change  the  origin  to  S  without  changing  the  direction 

of  either  axis.  o 

p    Zp 
The  co-ordinates  of  8  are  — 2'-~  (§  116).     Therefore  the 

nv   m    ^  ^ 


new  equation  is 


or 


y+l-T-^/^+i?'- 


my*  -f  ^.py  =  ^pmx. 


(1) 


THE    TARABOLA.  141 

Now  retain  the  axis  of  x,  and  turn  the  axis  of  y  till  it  coin- 
cides with  the  tangent  at  S ;  then  for  any  point  JP  we  have 

The  old  X  =  SB.  The  new  x  =  SJV. 

The  old  y  =  FJR.         The  new  y  =  JVF. 

And  it  is  easily  seen  from  Fig.  52  that 

SB  =  SJV+NF  cos  0, 
Pi?--iYPsin^. 

Therefore  equation  (1)  is  transferred  to  the  new  system  by 
writing  aj  +  ycos^  in  place  of  x,  and  y  sin  ^  in  place  of  y. 
Making  this  substitution,  and  reducing,  we  obtain 

y'=i^-.  (2) 

an  equation  of  the  same  form  as  y^  =  4:2?x. 
Join  /S  to  the  focus  F;  then 

Therefore  equation  (2)  may  be  more  simply  written 

y-  =  4;y.r,  (3) 

where  ^:»'  is  the  distance  of  the  origin  from  the  focus.  It  is 
easy  to  see  that  this  equation  includes  the  case  where  the 
axes  are  the  axis  of  the  curve  and  the  tangent  at  the  vertex. 

The  quantity  4^>'  is  called  the  Parameter  of  the  diameter 
passing  through  8.  When  the  diameter  is  the  axis  of  the 
curve,  it  is  called  the  Principal  Parameter. 

126.  Let  the  equation  of  a  parabola  referred  to  any  diam- 
eter, arid  the  tangent  at  the  end  of  that  diameter  as  axes, 
be  y^  =  ^jjK-c.  Since  the  investigations  in  §§  109-112  hold 
good  whether  the  axes  are  at  right  angles  or  not,  it  follows 
immediately  that  the  equation  of  the  tangent  at  any  point 
(a^i,  yi)    is    y{y  =  2j)\x-^x^,    and    that    the    straight    line 

y  =  onx  4-  —  will  touch  the  parabola  for  all  values  of  m. 


142 


ANALYTIC    GEOMETRY. 


127.    To  find  the  polar  equation  of  a  parabola,  the  focus 
being  the  pole. 


Fig.  53. 


Let  P  (Fig.  53)  be  any  point  (p,  6)  of  the  curve,  and  let  6 
be  measured  from  tbe  vertex  A  of  the  curve  in  the  same 
direction  as  clock  motion.     By  definition, 

PN^DM. 


Now 


Therefore 


or 


FP- 
FP  =  p, 

D2r=DF-MF, 
=  2^;  —  p  cos  0. 
p  =  2p  ~  p  cos  Oj 


P  = 


P 


1  4-  cos  e 


[29] 


THE    PARABOLA.  143 

Ex.  31. 

1.  Prove  that  the  polar  of  the  focus  is  the  directrix. 

2.  Prove  that  the  perpendicular  dropped  from  any  point 
of  the  directrix  to  the  polar  of  the  point  passes  through  the 
focus. 

3.  To  find  by  construction  the  pole  of  a  focal  chord. 

4.  Prove  that  through  any  point  tliree  normals  can  be 
drawn  to  a  parabola. 

5.  Tangents  are  drawn  through  the  ends  of  a  chord. 
Prove  that  the  part  of  the  corresponding  diameter  contained 
between  the  chord  and  the  intersection  of  the  tangents  is 
bisected  by  the  curve. 

6.  Focal  radii  are  drawn  to  two  points  of  a  parabola,  and 
tangents  are  then  drawn  through  these  points.  Prove  that 
the  angle  between  the  tangents  is  equal  to  half  the  angle 
between  the  focal  lines. 

7.  Prove  that  the  locus  of  the  intersection  of  two  tangents 
to  the  parabola  y^  =  ^px,  which  make  an  angle  of  45°,  is  the 
parabola  y'^  =  :r^  +  %px  -\-p^. 

8.  Explain  how  tangents  to  a  parabola  may  be  drawn  from 
an  exterior  point  (§§  121,  122). 

9.  Having  given  a  parabola,  how  would  you  find  its  axis, 
directrix,  focus,  and  latus  rectum  ? 

10.  From  the  point  (—  2,  5)  tangents  are  drawn  to  the 
parabola  y^^^a;.     What   is   the    equation   of  the   chord  of 

contact? 

11.  The  general  equation  of  a  system  of  parallel  chords  in 
the  parabola  7?/^  =  25.r  is  4:r  —  7?/-f  Z:  =  0.  What  is  the 
equation  of  the  corresponding  diameter  ? 


144  ANALYTIC    GEOMETRY. 

12.  In  the  parabola  i/  =  \?>x,  what  is  the  equation  of  the 
ordinates  of  the  diameter  ?/  +11=  0  ? 

13.  In  the  parabola  y'^=  6.t,  what  chord  is  bisected  at  the 
point  (4,  3)  ? 

14.  Given  the  parabola  y'^  =  4ipx]  find  the  equation  of  the 
chord  which  passes  through  the  vertex  and  is  bisected  by  the 
diameter  y  =  a.     How  can  this  chord  be  constructed  ? 

15.  The  latus  rectum  of  a  parabola  =  16.  What  is  the 
equation  of  the  curve  if  a  diameter  at  the  distance  12  from 
the  focus,  and  the  tangent  through  its  extremity,  are  taken  as 
axes? 

16.  Show  that  the  equations  of  that  chord  of  the  parabola 
y2  —  ^pr^  which  is  bisected  at  the  point  (A,  k)  is 

]c{jj  _  Jc)  =  2p(x  -  h). 

17.  Prove  that  the  parameter  of  any  diameter  is  equal  to 
the  double  ordinate  which  passes  through  the  focus. 

18.  Discuss  the  form  of  the  parabola  from  its  polar  equation. 

19.  Show  that  if  the  vertex  is  taken  as  pole,  the  polar  equa- 
tion of  a  parabola  is  4^5  cos^ 

^^    sin'^6>   * 

20.  Find  the  locus  of  the  foot  of  a  perpendicular  dropped 
from  the  focus  to  the  normal  to  a  parabola. 

21.  Two  normals  to  a  parabola  are  perpendicular  to  each 
other  ;  find  the  locus  of  their  intersection. 

22.  Find  the  locus  of  the  centre  of  a  circle  which  touches  a 
given  circle  and  also  a  given  straight  line. 

23.  The  area  and  base  of  a  triangle  being  given,  find  the 
locus  of  the  intersection  of  perpendiculars  dropped  from  the 
ends  of  the  base  to  the  opposite  sides. 


CHAPTER   VI. 

THE    ELLIPSE. 

Simple  Properties  of  the  Ellipse. 

128.  The  Ellipse  is  the  locus  of  a  point,  the  sum  of  whose 
distances  from  two  fixed  points  is  constant. 

The  fixed  points  are  called  Foci ;  and  the  distance  from  any 
point  of  the  curve  to  a  focus  is  called  a  Focal  Eadius. 

The  constant  sum  is  denoted  by  2  a,  and  the  distance  be- 
tween the  foci  by  2  c. 

Q 

The  fraction  -  is  called  the  Eccentricity,  and  is  represented 

by  the  letter  e.     Therefore  c  =  ae. 

From  the  definition  of  the  ellipse  it  is  clear  that  if  2a  <  2c, 
or  a  <  (?,  the  locus  does  not  exist ;  if  a  =  c,  the  locus  is  simply 
that  part  of  the  straight  line  joining  the  foci  which  is  com- 
prised between  the  foci.  The  ellipse  is  the  curve  obtained 
when  a'>  c  \  that  is,  when  e <  1. 

129.  To  construct  an  ellipse,  having  given  the  foci  and  the 
co'ostant  sum  2  a. 

I.  By  Motion.  Fix  pins  in  the  paper  at  the  foci.  Tie  a 
string  to  them,  making  the  length  of  the  string  exactly  equal 
to  2  a.  Then  press  a  pencil  against  the  string  so  as  to  make 
it  tense,  and  move  the  pencil,  keeping  the  string  constantly 
stretched.  The  point  of  the  pencil  will  trace  the  required 
ellipse  ;  for  in  every  position  the  sum  of  the  distances  from 
the  point  of  the  pencil  to  the  foci  is  equal  to  the  length  of  the 
string. 


146 


ANALYTIC    GEOMETRY. 


II.    By  Points.     Let  F,  F'  be  the  foci ;  then  FF^  =  2c. 

Bisect  FF'  at  0,  and  from  0  lay  off  OA  =  OA^  =  a. 

Then     AA'  =  2a. 

AF  +AF'  =-  (a-  c)  -{-  2c  -{-  (a-  c)  =  2a. 
AF'  +  A'F'  =  (a  -  c)  +  2c  +  (a  -  c)  =  2a. 

Therefore  A  and  A'  are  points  of  the  curve. 

Between  A  and  A'  mark  any  point  J^;  then  describe  two 
arcs,  one  with  i^  as  centre  and  ^Xas  radius,  the  other  with 
F'  as  centre  and  A'X  as  radius :  the  intersections  F,  Q  of 


these  arcs  are  points  of  the  curve.     By  merely  interchanging 
the  radii,  two  more  points,  F,  jS,  may  be  found. 

After  a  sufficient  number  of  points  has  been  obtained,  draw 
a  continuous  curve  through  them. 

130.  The  line  AA'  is  the  Transverse  or  Major  Axis,  A,  A' 
the  Vertices,  and  0  the  Centre  of  the  curve. 

The  line  FB',  perpendicular  to  the  major  axis  at  0,  is  the 
Conjugate  or  Minor  Axis  ;  its  length  is  denoted  by  2  b. 

Show  that  £  and  B'  are  equidistant  from  the  foci,  that 
£F=  a,  that  BO  =  h,  and  that  a'  =  b'  +  c\^ 


THE    ELLIPSE.  147 

131.    To  find  the  equation  of  the  ellipse,  having  given  the 
foci  and  the  constant  sum  2a. 


Take  the  line  AA^  (Fig.  55),  passing  through  the  foci,  as 
the  axis  of  x,  and  the  point  0,  half  way  between  the  foci,  as 
origin.  Let  P  be  any  point  {x,y)  of  the  curve,  and  let  r,  r' 
denote  the  focal  radii  of  F.  Then  from  the  definition  of  the 
curve,  and  from  the  right  triangles  FPM,  F^PM, 

T^'  =  y'^{c-{-xf  (1) 

,^  =  n/A^{c~xY  (2) 

By  addition,  r'-  +  r  =  2{x'  +  y'  +  c').  (3) 

By  subtraction,  r'"  —  r^  =  4:cx.  (4) 

Factor  (4),        (r'  +  r)  (r'-r)  =  4:  ex.  (5) 

Put  2  a  for  r'  +  r,  r'  -  r  -  —  (6) 

ex 

Whence  r  =  a ^  =  a  —  ex.  (7) 

a  ^  -' 

ex 
r'  =  a  +  —  =  a  4-  ex.  (8) 

a' +  ^^^  =  2(x' +  ,/  +  €')•  (9) 

Reduce,  and  substitute  h"^  in  place  of  a^  —  c"^  (§  130), 
h'x' +  ay  =  a'b', 

a^  +  iy.  =  ^'  [30] 


Substitute  in  (3), 


148  ANALYTIC    GEOMETRY. 

132.    To  trace  the  form  of  the  curve  from  its  equation. 

The  intercepts  on  the  axis  of  x  are  +  a  and  —  a ;  on  the 
axis  of  y,  -\-h  and  —  h. 

Only  the  squares  of  the  variables  x  and  y  appear  in  the 
equation  ;  hence,  if  it  is  satisfied  by  a  point  {x,  y),  it  will  also 
be  satisfied  by  the  points  {x,  —y),  (— ^,  y),  (— ^,  —y)-  There- 
fore we  infer  that 

(i.)    The  curve  is  symrnetrical  loith  respect  to  the  axis  of  x. 
(ii.)    The  curve  is  symmetrical  with  respect  to  the  axis  of  y. 
(iii.)  Every  chord  which  passes  through  the  point  0  is  bisected 
at  0 ;  for  the  distance  from  either  (:r,  y)  or  {—x,  —y)  to  (0,  0) 
is  ^x^-\-y'\     This  explains  why  0  is  called  the  centre. 

x\       ,    A/^^ 


Since  the  sum  of  f  -  J  and  (  't  j  is  1,  neither  of  these  squares 

can  exceed  1  ;  therefore  the  maximum  value  of  x  is  -j-  a,  and 
the  minimum  value  —  a,  while  the  corresponding  values  of  y 
are  -f  b  and  —  b.  Therefore  the  curve  is  wholly  contained 
within  the  rectangle  whose  sides  are  equal  to  2a  and  2b,  and 
are  bisected  by  the  axes. 

133.  To  trace  the  changes  in  the  form  of  the  ellipse  when  the 
semi-axes  are  supposed  to  change. 

Let  a  be  regarded  as  a  constant,  and  5  as  a  variable. 

(i.)  Suppose  b  to  increase.  Then  c  decreases  (since  c'  =  a^ 
—  h^),  e  decreases,  the  foci  approach  the  centre,  and  the  ellipse 
approaches  the  circle. 

(ii.)  Let  b  =  a.  Then  c  =  0,  e  =  0,  the  foci  coincide  with 
the  centre,  the  ellipse  becomes  a  circle  of  radius  a,  and  equa- 
tion [301  becomes 

Therefore  we  may  regard  a  circle  as  an  ellipse  whose  eccen- 
tricity is  equal  to  0. 


THE    ELLIPSE.  149 

(iii.)  Let  5  >  «.  The  foci  and  major  axis  will  now  be  on  the 
axis  of  I/,  c  will  increase  with  b,  e=  -,  and  b^  =  a'  -j-  c^. 

(iv.)  If  we  suppose  b  to  decrease  to  0  (a  remaining  con- 
stant), c  will  increase  to  a,  c  will  increase  to  1,  while  the 
curve  will  approach,  and  finally  coincide  with,  the  major  axis, 
its  equation  at  the  same  time  becoming  y  =  0. 

134.  It  follows  from  §  131  that  a  point  (A,  h)  is  on  the 
ellipse  represented  by  equation  [30],  provided 

-  +  --1  =  0. 
d'  ^b' 

It  may  be  shown  by  reasoning  similar  to  that  employed  in 
§  105  that  the  jDoint  {h,  Jc)  is  outside  or  inside  the  curve, 

according  as  —  +  7-,  —  1  is  positive  or  negative. 
d^      b' 

135.  Since  the  constants  a  and  b  in  the  equation 

may  have  any  positive  values,  every  equation  reducible  to  the 
form  of  (1)  must  represent  an  ellipse.  Hence  every  equation 
of  the  form  Ax^  +  Bf--=C 

represents  an  ellipse,  provided  C  is  not  zero,  and  A,  B,  and  C 
all  have  the  same  sign.     Its  semi-axes  have  the  values 


"=>iz  *=V: 


1^. 

136.    The  chord  passing  through  either  focus  perpendicular 
to  the  major  axis  is  called  the  Latus  Eectum  or  Parameter. 
To  find  its  length,  put  :r  =  c  in  the  equation  of  the  ellipse. 

Then  ,/=^lui'-c')=-,    y  =  d= -• 

-^       a^^  ^      a^     ^  a 

Therefore  the  latus  rectum  = 

a 


150 


ANALYTIC    GEOMETRY. 


137.  The  circle  having  for  diameter  the  major  axis  of  the 
ellipse  is  called  the  Auxiliary  Circle ;  its  equation  is 

x'  +  y'  =  a\ 

The  circle  having  for  diameter  the  minor  axis  is  called  the 
Minor  Auxiliary  Circle  ;  its  equation  is 

x'  +  y'^h'. 

If  P  (Fig.  56)  is  any  point  of  an  ellipse,  and  the  ordinate 
MP  produced  meets  the  auxiliary  circle  in  Q,  the  point  Q  is 
said  to  correspond  to  the  point  P. 

The  angle  QOM'i^  called  the  Eccentric  Angle  of  the  point  P, 
and  denoted  by  the  letter  <^. 


138.  Let  y,  y'  represent  the  ordinates  of  points  in  an  ellipse 
and  the  auxiliary  circle  respectively,  corresponding  to  the 
same  abscissa  x.     Then  from  the  equations  of  the  two  curves 

we  have  7     

y  =  -Va^  — ^^     ?/'  =  Va^  —  x\ 

Whence  y  -.y^  —  h  -.a,  or 

The  ordinates  of  the  ellipse  and  the  auxiliary  circle,  corre- 
sponding to  a  common  abscissa,  are  to  each  other  in  the  constant 
ratio  of  the  semi-axes  of  the  ellipse. 


THE    ELLIPSE.  151 

139.  Hence,  if  the  axes  2a,  2 i  of  an  ellipse  are  given,  we 
may  find  any  number  of  points  in  the  ellipse  by  constructing 
the  auxiliary  circle,  drawing  ordinates  at  pleasure,  and  then 
reducing  their  lengths  in  the  ratio  b  :  a. 

In  practice,  it  is  convenient  to  proceed  as  follows  : 
Construct  both  the  major  and  minor  auxiliary  circles;  draw 
any  radius,  cutting  the  circles  in  Q,  B,  respectively  ;  through 
Q  draw  a  line  II  to  0  F,  and  through  R  draw  a  line  li  to  OX: 
the  intersection  P  of  these  parallels  is  a  point  of  the  ellipse. 
For  from  the  similar  triangles  QOM,  ORN, 

ON:QM  ^OR:OQ. 

Now  •     ON^PM=y,         QM=y\ 

OR  =  b,         OQ  =a. 

Therefore  y :  y'  =  b  :  a. 

With  the  aid  of  the  eccentric  angle  (f>  --  QOX,  the  proof 
that  P  is  a  point  of  the  ellipse  may  be  given  as  follows  : 
Let  P  be  the  point  (x,  y)  ;  then 


X  =  0M=  OQ  coscfi^a  cos  <^, 

y  =  PM^-  0N=  OR  sin<f>^b  sin  0. 

Whence 

we  have 

X                    y        . 
-  =  cos  <f),     T  —  sm  d>. 
a            ^      b            ^ 

Square 

X^                                  7/2 

-2  =  cos^<^,     '^-sin^t^. 

Add 

X^          7/2 

^i  +  ^.  =  cos^<^  +  sin2<^. 

But 

cos^<^  +  sin^  cf>  =  l. 

Hence 

Therefore  P  is  a  point  of  the  required  ellipse. 


152 


ANALYTIC    GEOMETRY. 


140.  Another  mode  of  constructing  an  ellipse  from  its  axes 
is  shown  in  Fig.  57. 

In  the  rectangle  OACB,  whose  sides  OA,  OB  are  made 
equal  to  the  given  semi-axes  a  and  6,  divide  the  side  BC 
into  any  number  of  equal  parts,  and  divide  BO  into  the  same 
number  of  equal  parts,  and  let  31,  N  denote  any  two  corre-.. 
spending  points  of  division,  counting  from  B.  If  we  now  draw 
through  the  extremities  A,  A'  of  the  major  axis,  and  the  points 
M,  iV,  respectively,  straight  lines,  the  intersection  P  of  the 
lines  will  be  a  point  of  the  required  ellipse. 


In  order  to  give  a  general  proof,  let  there  be  n  equal  parts, 
and  let  ON  and  CM  contain  r  of  these  parts,  respectively  ; 
then  7 

n  n 

Produce  3INio  meet  OB  produced  in  Q;  then 

0Q:AC=0A:C3f, 

OQ:      h-. 

nh 


ra 
a  :  —  =  n  :  T. 
n 


Therefore 


0Q  = 


2+ 

a 

ry ^ 
nb 

-1. 

1+ 

ny 

-1. 

=  1- 

X 

a 

ny 

-'+1- 

--1- 

=  1. 

THE    ELLIPSE.  153 

Taking  now  0  for  origin,  and  OA  for  axis  of  x,  we  have  for 
the  symmetrical  equations  of  AM  and  A'JY 


and 
Or 

and 

Multiply 

that  is, 

This  relation  must  hold  true  of  the  point  common  to  the 
lines  J. J/  and  A'JV;  therefore  this  point  is  on  the  ellipse 
whose  axes  are  2a  and  2b. 

Ex.  32. 

What  are  a,  b,  c,  and  e  in  the  ellipse  whose  equation  is 

■  25      16 

2.  x-  +  2i/  =  2? 

3.  30,- +  4/ =  12? 

4.  Ax'+By'  =  l? 

5.  Find  the  latus  rectum  of  the  ellipse  Sx^  -{-  7?/^  =  18. 

6.  Find  the  eccentricity  of  an  ellipse  if  its  latus  rectum  is 
equal  to  one-half  its  minor  axis. 


154  ANALYTIC    GEOMETRY. 

What  is  the  equation  of  an  ellipse  if 

7.  The  axes  are  12  and  8  ? 

8.  Major  axis  =  2G,  distance  between  foci  =  24  ? 

9.  Sum  of  axes  =  54,  distance  between  foci  =  18  ? 

10.  Latus  rectum  =  -%*-,  eccentricity  =  |  ? 

11.  Minor  axis  =  10,  distance  from  focus  to  vertex  =  1  ? 

12.  The  curve  j^asses  through  (1,  4)  and  (—6,  1)  ? 

13.  Major  axis  =  20,  minor  axis  =  distance  between  foci? 

14.  Sum  of  the  focal  radii  of  a  point  in  the  curve  =  3  times 
the  distance  between  the  foci  ? 

15.  Prove  that  the  semi-minor  axis  is  a  mean  proportional 
between  the  segments  of  the  major  axis  made  by  one  of  the 
foci. 

16.  What  is  the  ratio  of  the  two  axes  if  the  centre  and  foci 
divide  the  major  axis  into  four  equal  parts  ? 

17.  For  what  point  of  an  ellipse  is  the  abscissa  equal  to  the 
ordinate  ? 

Find  the  intersections  of  the  loci 

18.  Sx'  +  6f  =  ll    and   7/  =  x  +  l. 

19.  2x''-\-3f  =  U   and   /  =  4.r. 

20.  x'-\-7f  =  lQ   and   x'-{-f  =  lO. 

21.  The  ordinates  of  the  circle  x'^-{-y^=r'^  are  bisected; 
find  the  locus  of  the  points  of  bisection. 

22.  A  straight  line  AB  so  moves  that  the  points  A  and  B 
always  touch  two  fixed  perpendicular  straight  lines.  Show 
that  any  point  JP  in  AB  describes  an  ellipse,  and  find  its 
equation. 


THE   ELLIPSE.  155 

23.  What  is  the  locus  of  Ax^-\-By-  —  C  when  C  is  zero? 
When  is  this  locus  imaginary  ? 

24.  Prove  that  the  abscissas  of  the  ellipse  h-oy  +  d'lf  =  a^b'' 

are  to  the  corresponding  abscissas  of  the  minor  auxiliary  circle, 
X'^  -j-  y2  _  ^2^  j^g  ^^ .  ^ 

25.  Construct  an  ellipse  by  the  method  of  §  139. 

26.  Construct  an  ellipse  by  the  method  of  §  140. 

27.  Construct  the  axes  of  an  ellipse,  having  given  the  foci 
and  one  point  of  the  curve. 

28.  Construct  the  minor  axis  and  foci,  having  given  the 
major  axis  (in  magnitude  and  position)  and  one  point  of  the 
curve. 

29.  A  square  is  inscribed  in  the  ellipse 

-4-^  =  1 
a'  ^  b' 

Find  the  equations  of  the  sides  and  the  area  of  the  square. 

Tangents  and  Normals. 

141.  To  find  the  equations  of  a  tangent  and  a  normal  to  an 
ellipse,  having  given  the  point  of  contact  {x^,  3/1). 

Taking  the  equation  of  the  ellipse, 
Z/V  +  ay  =  a^b\ 
and   the  equation   of  the   straight  line   through  (^1,  yi)   and 
(^^'2, 3/2) ,  y  —  7/1  _  ?/,  —  ?/i 

and  proceeding  as  in   §   72,  we  obtain  as  the   equation  of  a 
chord  through  (xi,  t/j)  and  (X2,  y^ 

x-Xi  a\yi  +  y,) 


156  ANALYTIC    GEOMETRY. 

Now  make  X2  =  Xi,  y^  =  yi  ;  then  the  chord  becomes  a  tan- 


becomes 


which  reduces  to        ^  +  ^  =  1.  [311 

From  the  equation  above  it  appears  that  the  value  of  the 
slope  of  the  tangent,  in  terms  of  the  co-ordinates  of  the  point 
of  contact  is  72 

0  Xi 

The  normal  is  perpendicular  to  the  tangent,  and  passes 
through  (xi,  1/1)  ;  therefore  its  equation  is  easily  found  (by  the 
method  of  §  51)  to  be 


h-         cC^  cc-lf- 


[32] 


142.    To  find  the  suhtangent  and  subnormal. 


Making  y  =  0  in  [31]  and  [32],  and  then  solving  the  equa- 
tions for  X,  we  obtain  : 

d^ 
Intercept  of  tangent  on  axis  of  ^  =  — > 

Xi 

Intercept  of  normal  on  axis  of  a;  =  —  ^1  =  e'Xi. 

d 

AVhence  the  values  of  the  subtangent  and  the  subnormal 
(defined  as  in  §  71)  are  easily  found  to  be  as  follows  : 

Subtangent  =  ^  ~  ^'\  [33] 

Subnormal  =  -~^\'  [34] 


THE    ELLIPSE. 


157 


143.  Jf  tangents  to  ellipses  having  a  common  major  axis  are 
draicn  at  points  having  a  common  abscissa,  they  will  vieet  on 
the  axis  of  x. 

For  in  all  these  ellipses  the  values  of  a  and  x  are  constant, 
and  therefore  (by  §  142)  the  tangents  all  cut  the  same  inter- 
cept from  the  axis  of  x. 


Fig.   58. 

144.    The  normal  at  any  point  of  an  ellipse  bisects  the  angle 
formed  by  the  focal  radii. 

The  values  of  the  focal  radii  for  the  point  P  (Fig.  58)  were 
found  in  §  131  to  be 

PF=  a  —  exi,     FF'  =  a-\-  ex^. 


(§  142)  ;  and  therefore 

NF  =  c  —  e^Xi  =  ae  —  e^Xi  -=  e{a  —  ex), 
JVF'  —  c-{-  e^Xi  =  ae-\-  e^x^  -~  e(a  +  ex). 

Therefore     NF :  NF^  ^  PF :  PF\ 

or  the  normal  divides  the  side  i^F'  of  the  A  PFF'  into  two 
parts  proportional  to  the  other  two  sides.  Therefore  (by- 
Geometry)  FPN=FPN. 

The  tangent  FT,  being  perpendicular  to  the  normal,  must 
bisect  the  angle  FFFi,,  formed  by  one  focal  radius  with  the 
other  produced. 


158 


ANALYTIC    GEOMETRY. 


145.  To  draiu  a  tangent  ayid  a  normal  through  a  given  point 
of  an  ellipse. 

I.  Let  P  (Fig.  59)  be  the  given  point.  Describe  the 
auxiliary  circle,  draw  the  ordinate  PM,  produce  it  to  meet 
the  circle  in  Q,  draw  Q 7^  tangent  to  the  circle  and  meeting 
the  axis  of  x  in  T,  and  join  PT\  then  PT  is  a  tangent 
to  the  ellipse  (§  143).  Draw  PN  J_  to  PT\  PN  is  the 
normal  at  P. 

a 


11.    Draw  the  focal  radii,  and  bisect  the  angles  between 
em.     The  bise( 
point  P  (§  144). 


them.     The  bisectors  are  the  tangent  and  the  normal  at  the 


146.    To  find  the  equation  of  a  tangent  to  an  ellipse,  having 
given  its  direction. 

This  problem  may  be  solved  by  finding  under  what  condi- 
tion the  straight  line  ,  ., . 

will  touch  the  ellipse     Irx'  +  cry~  —  a'^b'^.  (2) 

Eliminating  y  from  (1)  and  (2),  and  then  solving  for  x,  we 
find  two  values  of  :r  :       ^ 

—  ma-c  -t  ah^m'-a-  -\-  b'^  —  c^ 


vra 


b' 


These  values  will  be  equal  if 


m'a'  -i-b'-c^  =  0,  or  c=±  -Vni'd'  +  b' 


THE    ELLIPSE.  159 

If  the  two  values  of  x  are  equal,  the  two  values  of  y  must 
also  be  equal  from  equation  (1). 

Therefore  the  two  points  in  which  the  ellipse  is  cut  by  the 
line  will  coincide  if  d?  =  =b  VmW  +  b'\ 

Hence  the  straight  line 

y  =  mx  ±  {m^a"'  +  h"-)  [35] 

will  touch  the  ellipse  for  all  values  of  m. 

Since  either  sign  may  be  given  to  the  radical,  it  follows  that 
two  tangents  may  be  drawn  to  an  ellipse  in  a  given  direction 
(determined  by  the  value  of  m). 

147.  To  find  the  locus  of  the  intersection  of  two  tangents  to 
an  ellipse  which  are  perpendicular  to  each  other. 

Let  the  equations  of  the  tangents  be 


y^mx  +Vm'V  +  ^',  (1) 

y  =  m'x  +  ^m'd'  +  b'\  (2) 

The  condition  to  be  satisfied  is 

1        -      1 

mm  —  —  1,  or   ni  = 

If  we  substitute  for  m'  in  equation  (2)  its  value  in  terms  of 
m,  the  equations  of  the  tangents  may  be  written 


y  —  7nx  =  VmV  +  ^^,  (3) 

7ny  -{-  x  =  Va'^  +  7n^b'^.  (4) 

The  co-ordinates,  x  and  ?/,  of  the  intersection  of  the  tangents 

satisfy  both  (3)  and  (4)  ;  but  before  we  can  find  the  constant 

relation  between  them  we  must  first  eliminate  the  variable  m. 

This  is  most  easily  done  by  adding  the  squares  of  the  two 

equations  ;  the  result  is 

(1  +  m')x'  +  (1  +  rn')y'  -  (1  +  7n')  (a'  +  b'), 
or  x'  +  /  -  a'  +  b\ 

The   required   locus   is   therefore   a   circle.      This   circle   is 
called  the  Director  Circle  of  the  ellipse. 


160  ANALYTIC    GEOMETRY. 

Ex.  33. 

1.  What  are  the  equations  of  the  tangent  and  the  normal 
to  the  ellipse  2a;^+3y  =  35  at  the  points  whose  abscissa  =  2? 

2.  What  are  the  equations  of  the  tangent  and  the  normal  to 
the  ellipse  4 :f'^ -f- 9 y' =  36  at  the  points  whose  abscissa  =  —  f? 

3.  Find  the  equations  of  the  tangent  and  the  normal  to  the 
ellipse  x^  +  4:7/  =  20  at  the  point  of  contact  (2,  2).  Also  find 
the  subtangent  and  the  subnormal. 

4.  Show  that  the  line  7/  ^=  x  -\-  Vf  touches  the  ellipse 
2^'  +  3y^  =  l. 

5.  Required  the  condition  which  must  be  satisfied  in  order 
that   the   straight   line h  -  =  1    niay    touch    the    ellipse 

9  n  711/  77/ 

-4-^  =  1 

6.  In  an  ellipse  the  subtangent  for  the  point  (3,  -^-^)  is  16, 
the  eccentricity  =  4.     What  is  the  equation  of  the  ellipse  ? 

7.  AVhat  is  the  equation  of  a  tangent  to  the  ellipse 
9^;'  + 64?/' =--576  parallel  to  the  line  27/  =  x? 

8.  Find  the  equation  of  a  tangent  to  the  ellipse  3.r'-f5y'=15 
parallel  to  the  line  4a;  —  ^?/  —  1  =  0. 

.     ^        "h 

9.  In  what  points  do  the  tangents  which  are  equally  inclined 

to  the  axes  touch  the  ellipse  &V  +  «y  =  «^^^  ? 

10.  Through  what  point  of  the  ellipse  h'^x'^  +  ay  =  n^b"^ 
must  a  tangent  and  a  normal  be  drawn  in  order  that  they 
may  form,  with  the  axis  of  x  as  base,  an  isosceles  triangle  ? 

11.  Through  a  point  of  the  ellipse  5V  +  «y  =  <^^^",  and 
the  corresponding  point  of  the  auxiliary  circle  x-  -\- if  =  a^ 
normals  are  drawn.     What  is  the  ratio  of  the  subnormals  ? 


THE    ELLIPSE.  161 

12.  For  what  points  of  the  ellipse  b'^x^~\-  anf  =  c^lP'  is  the 
subtangent  equal  numerically  to  the  abscissa  of  the  point  of 
contact? 

13.  Find  the  equations  of  tangents  drawn  from  the  point 
(3,  4)  to  the  ellipse  16a;''  +  25y'  =  400. 

14.  What  are  the  equations  of  the  tangents  drawn  through 
the  extremities  of  the  latera  recta  of  the  ellipse  ^x^^^y^—  36a^  ? 

15.  What  is  the  distance  from  the  centre  of  an  ellipse  to  a 
tangent  making  the  angle  <^  with  the  major  axis? 

16.  What  is  the  area  of  the  triangle  formed  by  the  tangent 
in  the  last  parabola  and  the  axes  of  co-ordinates  ? 

17.  From  the  point  where  the  auxiliary  circle  cuts  the 
minor  axis  produced  tangents  are  drawn  to  the  ellipse.  Find 
the  points  of  contact. 

18.  Prove  that  the  tangents  drawn  through  the  ends  of  a 
diameter  are  parallel. 

19.  Find  the  locus  of  the  foot  of  a  perpendicular  dropped 
from  the  locus  to  a  tangent. 

Ex.  34.     (Revie-w.) 

1.  Given  the  ellipse  Z^x"  + 100/  =  3600.  Find  the  equa- 
tions and  the  lengths  of  focal  radii  drawn  to  the  point  (8,  -i^). 

2.  Is  the  point  (2,  1)  within  or  without  the  ellipse 
o^-2_^3^2^;L2? 

Find  the  eccentricity  of  an  ellipse 

3.  If  the  equation  is  2x^  +  3y'  =  1. 

4.  If  the  angle  FBF'  =  90°  (see  Fig.  54). 

5.  If  LFR  is  the  latus  rectum  and  LOR  is  an  equilateral 
triangle  (i^  being  the  focus,  0  the  centre). 


1G2  ANALYTIC    GEOMETRY. 

Find  the  equations  of  tangents  to  an  ellipse 

6.  If  tliey  make  equal  intercepts  on  the  axes. 

7.  If  they  are  parallel  to  BF  (Fig.  54). 

8.  Which  are  parallel  to  the  line  -  +  y  =  1  (a  and  b  being 
the  semi-axes). 

9.  Find  the  equation  of  a  tangent  in  terms  of  the  eccentric 
angle  <j!>  of  the  point  of  contact. 

Find  the  distance  from  the  centre  of  an  ellipse  to 

10.  A  tangent  through  the  point  of  contact  (xi,  y^). 

11.  A  tangent  making  the  angle  ^  with  the  axis  of  x. 

12.  In  what  ratio  is  the  abscissa  of  a  point  divided  by  the 
normal  at  that  point  ? 

13.  At  the  point  (.Tj,  ?/i)  of  an  ellipse  a  normal  is  drawn. 
What  is  the  product  of  the  segments  into  which  it  divides  the 
major  axis? 

14.  Find  the  length  of  PiV^  (Fig.  58). 

15.  Determine  the  value  of  the  eccentric  angle  at  the  end 
of  the  latus  rectum. 

Prove  that  the  semi-minor  axis  h  of  an  ellipse  is  a  mean 
proportional  between 

16.  The  distances  from  the  foci  to  a  tangent. 

17.  A  normal  and  the  distance  from  the  centre  to  the  cor- 
responding tangent. 

Determine  and  describe  the  loci  of  the  following  points  : 

18.  The  middle  point  of  that  portion  of  a  tangent  contained 
between  the  tangents  drawn  through  the  vertices. 

19.  The  middle  point  of  a  perpendicular  dropped  from  a 
point  of  the  circle  (a:  —  of  -\-y^  =  r^  to  the  axis  of  y. 


THE   ELLIPSE.  163 

20.  The  middle  point  of  sl  chord  of  the  ellipse  iV+ay =a^Z>^ 
drawn  through  the  positive  end  of  the  minor  axis. 

21.  The  vertex  of  a  triangle  whose  base,  2  c,  and  sum  of 
the  other  sides,  25,  are  given. 

22.  The  vertex  of  a  triangle,  having  given  the  base  2n  and 
the  product  k  of  the  tangents  of  the  angles  at  the  base. 

23.  The  symmetrical  point  of  the  right-hand  focus  of  an 
ellipse  with  respect  to  a  tangent. 

SUPPLEMENTARY   PROPOSITIONS. 

148.  Two  tangents  can  he  drawn  to  an  ellipse  from  any 
point ;  and  they  will  be  real,  coincident,  or  imaginary,  accord- 
ing as  the  point  is  outside,  on,  or  inside  the  curve. 

If  the  tangent  y^^7nx-\-'\/wb'c^-\-lf  pass  through  the  point 

(A,/?:),  then  ;^  =.  ^i  +  V^>^V+F, 

or  {K'  -  a") iw  -  2  hhn  +  1^  -b'  =  0. 

This  is  a  quadratic  equation  with  respect  to  ni,  and  its  roots 
give  the  directions  of  those  tangents  which  pass  through  (h,  k). 
Since  a  quadratic  equation  has  tivo  roots,  two  tangents  may 
be  drawn  from  any  point  (A,  h)  to  an  ellipse. 

By  solving  the  equation,  w^e  obtain 


hk  ±  Vb'h:'  +  rr^'  -  a'b' . 

m  = > 

h'-a' 

and  we  see  that  the  roots  are  real,  coincident,  or  imaginary, 

according  as  b'h^  -f  ci^k"^  —  a'b^  is  positive,  zero,  or  negative  ; 

7^2      k'         .  .  ■ 

that  is,  according  as  —  4-77,-1  is  positive,  zero,  or  negative ; 

in  other  words,  according  as  the  point  {h,  k)  is  outside,  on, 
or  inside  the  ellipse.    (§  134.) 


164 


ANALYTIC    GEOMETRY. 


149.  To  find  the  equation  of  the  straight  line  passing  through 
the  points  of  contact  of  the  tiuo  tangents  drawn  to  an  ellipse  from 
the  'point  (A,  k). 

If  (^1, 2/1)  and  {x-i,  2/2)  are  the  points  of  contact,  it  follows, 

by   reasoning    similar    to    that   employed   in   §    78,   that  the 

required  equation  is 

hx      ky  _ 

a'  ~^  b'  ~  ^' 

This  line  is  always  real ;  but  if  the  point  (A,  Jc)  is  within 
the  ellipse,  the  points  (.ri,  yi),  {x2, 3/2)  through  which  the  line 
passes,  will  be  imaginary. 

150.  The  straight  line  joining  the  points  of  contact  of  the 
two  tangents  from  any  point  P  to  an  ellipse  is  called  the  Polar 
of  P  with  respect  to  the  ellipse  ;  and  the  point  P  is  called 
the  Pole  of  the  straight  line  with  respect  to  the  ellipse. 


Fig.   60. 


The  propositions  in  §§  80-82,  relating  to  poles  and  polars 
with  respect  to  a  circle,  also  hold  true  for  poles  and  polars 
with  respect  to  an  ellipse,  and  may  be  proved  in  the  same 
way. 


THE    ELLIPSE.  165 

151.  To  draw  a  tangent  to  an  ellipse  from  a  given  point  P 
outside  the  ciirve. 

Suppose  the  2:)roblem  solved,  and  let  the  tangent  meet  the 
ellipse  at  Q  (Fig.  60).  If  F^Q  be  produced  to  (7,  making 
QG  =  QF,  then  A  FQG  is  isosceles  ;  now  Z  FQF^-Z  GQF 
(§  144)  ;  therefore  FQ  is  perpendicular  to  FG  at  its  middle 
point ;  therefore  F  is  equidistant  from  F  and  G.  This  re- 
duces the  problem  to  determining  the  point  G. 

Since  F'G  =  2a,  G  lies  in  the  circle  with  F'  as  centre  and 
2  a  as  radius.  And  G  also  lies  in  the  circle  with  F  as  centre 
and  FF  as  radius.     Hence  the  construction  is  obvious. 

152.  To  find  the  locus  of  the  middle  points  of  the  sijstcm  of 
chords  represented  by  the  equation 

y  —  mx  +  Ic.  (1) 

Let  any  one  of  the  chords  meet  the  ellipse 

in  the  points  (:ri,  y^  and  (x^,  t/J  ;  then  (§  141  (3)) 

h\xy  +  a:,)  ,oN 

If  {x,y)  is  the  middle  point,  2x.^^  x^-\-  x^,  2y  =  y^~\-y^^ 
and  (3)  becomes  72 

m  =  -^,  (4) 

or  y  =  ___.  (5) 

a^m  -^ 

This  relation  holds  true  for  the  middle  points  of  all  the 
chords  ;  therefore  it  is  the  equation  required. 

The  locus  of  the  middle  points  of  a  system  of  parallel  chords 
in  an  ellipse  is  called  a  Diameter  of  the  ellipse. 

From  the  form  of  (5)  we  see  that  a  diameter  is  a  straight 
line  passing  through  the  centre. 


166  ANALYTIC    GEOMETRY. 

153.  Let  m'  denote  the  slope  of  the  diameter  of  the  chords 
represented  by  the  equation  y  =  77ix-j-k  ;  then  from  (5)  of  §  151 

mm'  =  —  ^*  [3G] 

From  the  symmetry  of  this  equation  we  may  infer  at  once 
that  all  chords  parallel  to  the  diameter  3/  =  7n'x  are  bisected 
by  the  diameter  1/  =  vix  ;  hence 

7/^  one  diameter  bisect  all  chords  'parcdlel  to  another,  the 
second  diameter  bisects  all  chords  parallel  to  the  first. 

Two  such  diameters  are  called  Conjugate  Diameters. 

154.  Let  a  straight  line  cutting  the  case  in  JP  and  Q  move 
parallel  to  itself  till  P  and  Q  coincide  with  the  end  of  the 
diameter  bisecting  PQ  ;  then  the  straight  line  becomes  the 
tangent  at  the  end  of  the  diameter.     Therefore 

A  diameter  bisects  all  chords  parallel  to  the  tangent  at  its 
extremity.  A  tangent  drawn  through  the  end  of  a  diameter 
is  parallel  to  the  conjugate  diameter  (§  153). 

155.  Let  POP\  EOPJ  (Fig.  61)  be  two  conjugate  diameters, 
meeting  the  curve  on  the  positive  side  of  the  axis  of  x  in  the 
points  P  {xi,  7/1)  and  P  {X2,  y^),  and  making  the  angles  a,  ^ 
respectively  with  the  axis  of  x. 

Let  a  be  acute  ;  then  it  follows  from  equation  [36]  that  ^ 
must  be  obtuse  ;  whence  we  infer  that  tivo  corijugate  diameters 
must  lie  in  different  quadrants. 

The  equation  of  the  tangent  through  P  is 

Therefore  the  equation  of  the  diameter  POP',  which  is 
parallel  to  this  tangent  (§  154)  and  passes  through  0,  is 

^  +  ^^  =  0.  (2) 


THE    ELLIPSE. 


167 


Similarly,  the  equation  of  FOF'  is  found  to  be 


^  4_  y-'^'/  _  n 


The  point  i?  is  in  the  locus  of  (2)  ;  therefore 
a'  ~^  b'    "^• 


(3) 


(4) 


Equation  (4)  is  the  condition  which  must  be  satisfied  by 
the  co-ordinates  of  the  extremities  of  every  pair  of  conjugate 
diameters. 


Fig.  61. 

156.  Let  the  ordinates  of  the  extremities  F,  R  (Fig.  61)  of 
two  conjugate  diameters  meet  the  auxiliary  circle  in  Q,  S 
respectively,  join  QO  and  SO,  and  denote  Z  QOX  by  ^, 
Z.  SOX  by  <ji\  Then  the  values  of  the  co-ordinates  of  F 
and  F  are  (§  138) 

Xi^=  a  cos  ^,         x.i-=  a  cos  <^', 

yi'=h  sin  <f>,         y-i  =  h  sin  ^'. 

Whence,  by  substitution  in  equation  (4)  of  §  155,  we  obtain 

cos  (^  cos  <^'  +  sin  <^  sin  <^'  ^=^  0. 
Therefore  <^'-<^  =  ^7r. 

That  is,  the  difference  of  the  eccentric  cingles  corresponding 
to  the  ends  of  two  conjugate  diameters  is  equal  to  a  right  angle. 


168  ANALYTIC   GEOMETRY. 

157.  Given  the  end  {x^,  y^)  of  a  diameter,  to  find  the  end 
{x.,,  T/a)  of  the  conjugate  diameter. 

From  §  156  we  have  for  one  of  the  ends 

rr^  =  a  cos  fp'  =  a  cos  (<f> -}-  ^  tt)  =  a  sin  <f), 

2/2  =  ^  sin  </)'  =  Z>  sin  (<f> -{-  ^  tt)  =  b  cos  tf). 

X2      —  a  sin  4>  a      ]).i      h  cos  ^      h 

yi         5  sin  ^    ~      V     x^  ~  a  cos  ^      a 

Therefore  x^~  —  -?/i,     y.^  =  -Xi. 

0  a 

Since  every  chord  through  the  centre  is  bisected  by  the 
centre,  the  co-ordinates  of  the  other  end  of  the  diameter  are 

a  1        b 

-?/i  and   —-x^. 
0  a 

158.  To  find  the  angle  formed  by  two  conjugate  semi-diam- 
eters, ivhose  lengths  a',  b'  are  given. 

Let  the  semi-diameters  make  the  angles  a,  /3  respectively 
with  the  axis  of  x,  and  let  0  denote  the  required  angle.  Then 
if  (xi,  2/1)  and  (x.2, 3/2)  are  the  extremities  of  a'  and  b'  respect- 


ively, 


sm  a  =  ^.      sm  p  =  -Tt  —  ^Tt' 
a''  '^       b'      ab' 

Xi  x\  ay^ 

cos  a  =  -,,       cos  p—~-r,  =  —  TTi' 

sin  6  =  sin  (/?  —  a) 

=  sin  ^  cos  tt  —  cos  /?  sin  a 

a6a'6' 
a^6^ 


a6a'6' 
ab 


THE    ELLIPSE.  169 

159.  The  lines  joining  any  point  of  an  ollipse  to  the  ends 
of  any  diameter  are  called  Supplemental  Chords. 

Let  PQ,  PQ  be  two  supplemental  chords  (Fig.  G2).  Through 
the  centre  0  draw  OR  parallel  to  PQ,  and  meeting  PQ  in  B  ; 
also  OR^  parallel  to  PQ,  and  meeting  P'Q  in  R'. 


Fig.  62. 


Since  0  is  the  middle  point  of  PP\  and  OR  is  drawn 
parallel  to  PQ,  and  OPJ  is  drawn  parallel  to  PQ,  R  and 
i?'  are  the  middle  points  of  QP,  QP  respectively.  Therefore 
OR  will  bisect  all  chords  parallel  to  QP,  and  OPJ  will  bisect 
all  chords  parallel  to  QP.  Hence  OR,  OP  are  conjugate 
diameters. 

Therefore  the  diameters  parallel  to  a  pair  of  supplemental 
chords  are  conjugate  diameters. 

160.  To  find  the  equation  of  an  ellipse  referred  to  a  pair  of 
conjugate  diameters  as  axes. 

The  origin  not  being  altered,  we  must  substitute  for  x  and  y 
expressions  of  the  form  ax  -\-  hy,  a'x  +  i>'y  (§  95). 

Therefore  the  transformed  equation  will  have  the  form 

Ax'-j-Cxy-]-£y'  =  l. 


170  ANALYTIC    GEOMETRY. 

But  by  hypothesis  the  axis  of  x  bisects  all  chords  parallel 
to  the  axis  of  y  ;  therefore  the  two  values  of  y  corresponding 
to  each  value  of  x  will  be  equal  and  of  opposite  signs. 

Hence  (7=  0,  and  the  equation 

Ax""  +  Cxy  +  ^3/2  =  1 

becomes  Ax""  +  Bij''  =  1 . 

The  intercepts  of  the  curve  on  the  new  axes  are  equal  to 
the  semi-conjugate  diameters.  If  we  denote  them  by  a'  and 
b',  we  have 


^i5  '-Vi 


whence  ^  =  — -.        B  =  -- » 

and  the  equation         ^^,2  1  ^^  ,2  ^  i 

becomes  a^  "^  6^  ~  P'''] 

This  equation  has  the  same  form  as  the  equation  referred 
to  the  axes  of  the  curve  ;  whence  it  follows  that  formulas 
derived  from  equation  [30],  by  processes  which  do  not  pre- 
suppose the  axes  of  co-ordinates  to  be  rectangular,  hold  true 
^vhen  we  employ  as  axes  two  conjugate  diameters. 

For  example,  the  ecjuation  of  a  tangent  at  the  point  (xi,  ?/i), 
referred  to  the  semi-conjugate  diameters  a'  and  b',  is 

161.  To  find  the  conditions  under  ivhich  an  equation  will 
represent  an  ellipse  when  of  the  form 

Ax"  +  By'^-^Dx  +  Ey-YF=  0. 

If  neither  A  nor  B  is  zero,  we  mav  wa-ite  the  equation 


THE    ELLIPSE.  171 

If  we  take  a  new  origin  with  parallel  axes  at  the  point 
(———-'  ~qd)'  ''^^^^  denote  by  A"  the  constant  quantity 
which  forms  the  right  side  of  the  last  equation,  the  equa- 
tion  becomes  Ax^+Bj/''=K, 

or  ^x'  +  -^7/=^l; 

x^       ?/^ 
that  is,  H'^'k^^' 


which  we  know  (§  135)  represents  an  ellipse,  provided  K  be 
not  zero  and  A,  B,  and  -ST have  like  signs.,,, 

K         K  *  ■ 

If  the  denominators  —  and  —  are  both  ncqative,  it  is  clear 
A  B  J         ^ 

that  no  real  values  of  x  and  y  will  satisfy  the  equation  :  the 

locus  in  this  case  is  called  an  imaginary  ellipse. 

It  is  obvious  that  the  two  denominators  —  and  —  will  have 

A  B 

like  signs  when  A  and  B  have  like  signs  ;  and  by  comparing 
the  signs  of  the  constants  which  enter  into  the  value  of  K,  it 
appears  that  the  common  sign  of  the  denominators  will  be 
positive  or  negative,  according  as  the  sign  of  F  is  unlike  or 
like  that  of  A  and  B.     Hence  an  equation  of  the  form 

-        Ax--^By''  +  Dx-\-Ey-{-F-=0 

(i.)  will  represent  an  ellipse,  if  A  and  B  are  neither  of  them 
zero  and  agree  in  sign. 

(ii.)  The  ellipse  will  he  real  or  imaginary,  according  as  the 
sign  of  F  is  unlike  or  like  that  of  A  and  B. 

The  axes  of  the  ellipse  are  parallel  to  the  axes  of  co-ordi- 

nates,  and  the  centre  is  the  point  f  —  — >  

^  V     --^        2i? 

The  major  axis  will  be  parallel  to  the  axis  of  x  or  to  the 
axis  of  y,  according  as  A  is  less  than,  or  greater  than,  B. 


172 


ANALYTIC    GEOMETRY. 


162.  To  find  the  locus  of  a  point  which  'moves  so  that  the 
ratio  of  its  distances  from  a  fixed  point  and  a  fixed  straight  line 
is  constant  and  less  than  unity. 


Let  e  denote  the  constant  ratio,  2p  the  distance  from  the 
fixed  point  i^(Fig.  63)  to  the  fixed  line  CE.  Taking  CE  for 
the  axis  of  y,  and  the  perpendicular  to  CE  through  F  for  the 
axis  of  X,  then  from  the  definition  of  the  locus 

FP=exNP=ex. 

Therefore  we  have  the  relation 

{x  —  2py-\-y'=e^x\ 

or  (l-e>'  +  y'-4;9;r  +  4/  =  (i  (1) 

Since  we  suppose  e<\,  the  coefficients  of  x""-  and  y"^  are 
both  positive  ;  therefore  the  locus  is  an  ellipse  (§  161).  Com- 
paring the  coefficients  of  (1)  with  those  of  the  first  equation 
of  (§  161),  we  obtain 

A  =  l-e\     B  =  l,     C=^-^p,     i)-0,     E=^p\ 

-p 


Therefore  the  centre  of  the  locus  is  the  point  f    ""-^   .^.  0 


THE   ELLIPSE.  173 

Changing  the  origin   to   the   centre,  we   obtain   an  equation 
which  may  be  written  in  the  form 

1   _  ./A  2  1   _  Ai 

)  x'  + -f  =  1.  (2) 

2cp)   ^{:icpr  ^^ 

By  putting  x  and  y  successively  equal  to  0,  we  find  for  the 
values  of  the  semi-axes 

2e»        ,  2e» 

a  =  z. — ^,,     b  =     , 

1  -  c'  Vi  -  e^ 

AVhence,  by  substitution  in  (2),  ^ve  get  the  equation  in  the 
ordinary  form  2        2 

1  +  1  =  1.  (3) 

Since  OD  =  -^   and  FB  ^-  2p,  therefore 

But  the  distance  from  the  centre  0  to  the  focus  of  the 
ellipse  is  o  2 

\  —  &■ 

Therefore  the  fixed  point  F  coincides  with  the  focus  of  the 

ellipse. 

Also  -'=  e,  or  the  constant  ratio  e  is  equal  to  the  eccen- 
a 

tricity  of  the  ellipse  as  defined  in  §  128. 
Whence  an  ellipse  is  often  defined  as 

The  locus  of  a  point  which  moves  so  that  the  ratio  of  its  dis- 
tances frovi  a  fixed  point  and  a  fixed  straight  line  is  constant 
and  less  than  unity. 

i^is  called  the  Focus  ;  FN,  the  Directrix. 

The  symmetry  of  the  curve  with  respect  to  the  minor  axis 
shows  that  there  is  another  focus  and  another  directrix  on  the 
other  side  of  the  minor  axis,  at  distances  from  it  equal  respec- 
tively to  those  of  F  and  CE. 


174 


ANALYTIC    GEOMETRY. 


163.    To  find  the  polar  equation  of  the  ellipse,  the  right-hand 
focus  being  taken  as  tJte  pole. 


Fig.  64. 

Let   P  be   any  poiDt   (p,  0)   of  the   curve.      Then   in   the 
triangle  PFF' 

FF''  ==^FF'  +  W  +  2PF X  FF^  XcosO. 

But  FF=  p,  FF'  =  2c,  and  by  the  definition  of  the  curve 
FF'  =  2a  —  p;  therefore 

(2a  -py  =  p''-i-4:c'-t4:cp  cos  6. 

Reducing,  and  substituting  aV  for  c-,  we  obtain 


P  = 


a'' 


•-     _   a(l  —  €-} 
a  +  c  cos  6  ~~  1  +  e  cos  e 


[38] 


We  may  obtain  this  result  more  simply  by  using  for  the 
focal  radius  FF  the  value  (§  131) 

p  =  a  —  ex. 

But  the  general  value  of  x  (0  being  acute)  is 

X  =  c  -\-  p  cos  0  =  ae  -]-  p  cos  0. 

By  substituting  this  value  of  x  we  obtain  the  same  polar 
equation  as  before. 


THE   ELLIPSE. 


175 


164.    To  find  the  area  of  an  ellipse. 

Divide  the  semi-major  axis  OA  (Fig.  65)  into  any  number 
of  equal  parts,  through  any  two  adjacent  points  of  division 
J/,  iV^  erect  ordinates,  and  let  the  ordinate  through  M  meet 


the  ellipse  in  P  and  the  auxiliary  circle  in  Q.  Through  P,  Q 
draw  parallels  to  the  axis  of  x,  meeting  the  other  ordinate  in 
P,  jS,  respectively.     Then  (§  138) 

area  of  rectangle  MPPN _  MP _  b^ 
area  of  rectangle  MQSN       MQ  ~  a 

And  a  similar  proportion  holds  true  for  every  correspond- 
ing pair  of  rectangles. 

Therefore,  by  the  Theory  of  Proportions, 

sum  of  rectangles  in  ellipse  _  h 
sum  of  rectangles  in  circle        a 

This  relation  holds  true  however  great  the  number  of  rect- 
angles. The  greater  their  number,  the  nearer  does  the  sum  of 
their  areas  approach  the  area  of  the  elliptic  quadrant  in  one 
case,  and  the  circular  quadrant  in  the  other.  In  other  words, 
these  two  quadrants  are  the  limits  of  the  sums  of  the  two 


176  ANALYTIC    GEOMETRY. 

series  of  rectangles.     Therefore,  by  the  fundamental  theorem 

of  limits,  r     ^^■    I-  i         i  7 

'  area  ot  elliptic   quadrant  _  o 

area  of  circular  quadrant       a 

Multiplying  both  terms  of  the  first  ratio  by  4, 

area  of  the  ellipse  _  h 
area  of  the  circle         a 

But  the  area  of  the  circle  =  tto^  ;  therefore 

area  of  the  ellipse  =  trab,  [39] 

Ex.  35. 

1.  "What  are  the  equations  of  the  directrices  (§  162)  ? 

2.  Prove  that  the  polars  of  the  foci  are  the  directrices. 

3.  What  is  the  equation  of  the  polar  of  the  point  (5,  7) 
with  respect  to  the  ellipse  Ax^  +  9?/^=  36  ? 

4.  Prove  that  a  focal  chord  is  perpendicular  to  the  line 
Avhich  joins  its  pole  to  the  focus.     In  what  line  does  the  pole 

lie? 

6.  Find  the  pole  of  the  line  Ax  -f-  Bi/  +(7=0  with  respect 
to  th5  ellipse  5V  -f  t«Y'^  =  a^^'^- 

6.  Each  of  the  two  tangents  which  can  be  drawn  to  an 
ellipse  from  any  point  on  its  directrix  subtends  a  right  angle 
at  the  focus. 

7.  The  two  tangents  which  can  be  drawn  to  an  ellipse  from 
any  external  point  subtend  equal  angles  at  the  focus. 

8.  Find  the  slope  vi  of  a  diameter  if  the  square  of  the 
diameter  is  (i.)  an  arithmetic,  (ii.)  a  geometric,  (iii.)  an  har- 
monic mean  between  the  squares  of  the  axes, 

9.  Given  the  length  2/  of  a  diameter,  its  inclination  0  to 
the  axes,  and  the  eccentricity  ;  find  the  major  and  minor  axes. 


THE    ELLirSE.  177 

10.  Tangents  are  drawn  from  (3,  2)  to  the  ellipse  a:^-f  4?/^~4. 
Find  the  equation  of  the  chord  of  contact,  and  of  the  line 
which  joins  (3,  2)  to  the  middle  point  of  the  chord. 

11.  Find  the  equation  of  a  diameter  parallel  to  the  normal 
at  the  point  (xi,  3/1),  the  semi-axes  being  a  and  b. 

12.  Find  the  area  of  the  rectangle  whose  sides  are  the  two 
segments  into  which  a  focal  chord  is  divided  by  the  focus. 

13.  What  is  the  equation  of  a  chord  in  the  ellipse 
13:r^  +  lly^  ~  1"13  which  passes  through  (1,2)  and  is  bisected 
by  the  diameter  3a:  —  2?/  =  0  ? 

14.  In  the  ellipse  9  x' +  SQ  f  =  324  find  the  equation  of  a 
chord  passing  through  (4,  2)  and  bisected  at  this  point. 

15.  Write  the  equations  of  diameters  conjugate  to  the  fol- 
lowing lines : 

X  —  1/  =  0,     X  -\-  ^  =  0,     ax  =  hi/,     ay  =  hx. 

16.  Show  that  the  lines  1x  —  y  =  0,  :r  -f  3  ?/  =  0  are  con- 
jugate diameters  in  the  ellipse  2a;^  -f  3y''^  =  4. 

17.  If  a',  V  are  two  semi-conjugate  diameters,  prove  that 

18.  The  area  of  the  parallelogram  formed  by  tangents 
drawn  through  the  ends  of  conjugate  diameters  is  constant, 
and  equal  to  4a6. 

19.  The  diagonals  of  the  parallelogram  in  No.  18  are  also 
conjugate  diameters. 

20.  The  angle  between  two  semi-conjugate  diameters  is  a 
minimum,  when  they  are  equal. 

21.  The  eccentric  angles  corresponding  to  equal  semi-con- 
jugate diameters  are  45°  and  135°. 

22.  The  polar  of  a  point  in  a  diameter  is  parallel  to  the 
conjugate  diameter. 


178  ANALYTIC    GEOMETRY. 

23.  Find  the  equations  of  equal  conjugate  diameters. 

24.  The  length  of  a  semi-diameter  is  I ;  find  the  equation 
of  the  conjugate  diameter. 

25.  The  angle  between  two  equal  conjugate  diameters  =: 
60°  ;  find  the  eccentricity  of  the  elli|)se. 

26.  Given  a  diameter,  to  construct  the  conjugate  diameter. 

27.  To  construct  two  conjugate  diameters  which  shall  con- 
tain a  given  angle. 

28.  To  draw  a  tangent  to  a  given  ellipse  parallel  to  a  given 
straight  line. 

29.  Given  an  ellipse  ;  to  find  by  construction  the  centre, 
foci,  and  axes. 

30.  Find  the  rectangular  equation  of  the  ellipse,  taking  the 
origin  at  the  left-hand  vertex. 

31.  Find  the  polar  equation  of  an  ellipse,  taking  as  pole 
the  left-hand  focus. 

32.  Find  the  polar  equation  of  the  ellipse,  taking  the  centre 
as  pole. 

33.  Discuss  the  form  of  the  ellipse  by  means  of  its  polar 
equation. 

34.  If  the  centre  of  an  ellipse  is  the  point  (4,  7),  and  the 
major  and  minor  axes  are  14  and  8,  find  its  equation,  the  axes 
being  supposed  parallel  to  the  axes  of  co-ordinates. 

35.  The  equation  of  an  ellipse,  the  origin  being  at  the  left- 
hand  vertex,  is  2b  x^  +  ^\y^  —  450:^  ;  find  the  axes. 

36.  If  the  minor  axis  =  12,  and  the  latus  rectum  =  5, 
what  is  the  equation  of  the  ellipse,  the  origin  being  taken  at 
the  left-hand  vertex  ? 


THE    ELLIPSE.  179 

Find  the  centre  and  axes  of  the  following  ellipses : 

37.  4.x'-{-i/-{-8x-2y  +  l  =  0. 

38.  9:i-"'  +  lGy^-3G.r-12Sy-f- 148  =  0. 

39.  4.r'^  +  36y-  +  36y  =  0. 

40.  Find  the  eccentric  angle  <^  corresponding  to  the  diam- 
eter whose  length  is  2  c. 

41.  At  the  intersection  of  the  ellipse  h'x'  +  «y  =  ci^l>''  and 
the  circle  x'^-j-y^  =  ah  tangents  are  drawn  to  both  curves. 
Find  the  angle  between  them. 

42.  How  would  you  draw  a  normal  to  an  ellipse  from  any 
point  in  the  minor  axis  ? 

43.'   Find  the  equation  of  a  chord  bisected  at  a  point  {h,  k). 

44.  Prove  that  the  length  of  a  line  drawn  from  the  centre 
to  a  tangent,  and  parallel  to  either  focal  radius  of  the  point 
of  contact,  is  equal  to  the  semi-major  axis. 

45.  A  circle  described  on  a  focal  radius  will  touch  the 
auxiliary  circle. 

46.  Find  the  locus  of  the  intersection  of  tangents  drawn 
through  the  ends  of  conjugate  diameters  of  an  ellipse. 

47.  Find  the  locus  of  the  middle  point  of  the  chord  joining 
the  ends  of  two  conjugate  diameters. 

48.  Find  the  locus  of  the  vertex  of  a  triangle  whose  base 
is  the  line  joining  the  foci,  and  whose  other  sides  are  parallel 
to  two  conjugate  diameters. 

49.  Find  the  locus  of  the  centre  of  a  circle  which  passes 
through  the  point  (0,  3)  and  touches  internally  the  circle 
x'^y'  =  2b. 


CHAPTER  VII. 
THE    HYPERBOLA. 

Simple  Properties  of  the  Hyperbola. 

165.  The  Hyperbola  is  the  locus  of  a  point  the  difFerence  of 
whose  distances  from  two  fixed  points  is  constant. 

The  fixed  points  are  called  the  Foci,  and  a  line  joining  any 
point  of  the  curve  to  a  focus  is  called  a  Focal  Eadius. 

The  constant  difFerence  is  denoted  by  2  a,  and  the  distance 
between  the  foci  by  2(?. 

The  fraction  -  is  called  the  Eccentricity,  and  is  denoted  by 
a 

the  letter  e.     Therefore  c  =  etc. 

Since  the  difFerence  of  two  sides  of  a  triangle  is  always  less 
than  the  third  side,  we  must  have  in  the  hyperbola 

2a  <  2(7,  or  a  <  c,  or  e>l. 

166.  To  construct  an  hyperbola,  having  given  the  foci,  and 
the  constant  difference  2  a. 

I.  By  Motion  (Fig.  66).  Fasten  one  end  of  a  ruler  to  one 
focus  jP'  so  that  it  can  turn  freely  about  F'.  To  the  other 
end  fasten  a  string.  Make  the  length  of  the  string  less  than 
that  of  the  ruler  by  2  a,  and  fasten  the  free  end  to  the  focus  F. 
Press  the  string  against  the  ruler  by  a  pencil  point  P,  and 
turn  the  ruler  about  F'. 

The  point  F  will  describe  one  branch  of  an  hyperbola. 
The  other  branch  may  be  described  in  the  same  way  by  inter- 
changing the  fixed  ends  of  the  ruler  and  the  string. 


THE    IIYrEIlBOLA. 


181 


11.    By  Foinis  (Fig.  67).     Let  F,  F'  be  the  foci ;  then 

FF'=2.c 
Bisect  FF'  at  0,  and  from  0  hiv  off  OA  =--  OA'  =  a. 


Then, 


AA' 


AF'-AF  =2a  +  (c-a)-(c-a)  =  2a. 
A'F-A'F'=^2a-i-lc-a)-lc-a)  =  2a. 
Therefore  A  and  A'  are  points  of  the  curve. 


Fig.  66. 


Fig.  67. 


In  AA'  produced  mark  any  point  F ;  then  describe  two 
arcs,  the  first  with  F  as  centre  and  AF  as  radius,  the  second 
with  F'  as  centre  and  A'F  as  radius;  the  intersections  F,  Q 
of  these  arcs  are  points  of  the  curve.  By  merely  interchang- 
ing the  radii,  two  more  points  F,  S  may  be  found. 

Proceed  in  this  way  till  a  sufficient  number  of  points  has 
been  obtained ;    then  draw  a  smooth  curve  through  them. 

Through  0  draw  FF'  _L  to  FF' ;  since  the  difference  of 
the  distances  of  every  point  in  the  line  FF'  from  the  foci  is 
0,  therefore  the  curve  cannot  cut  the  line  FF'. 

The  locus  evidently  consists  of  two  entirely  distinct  parts  or 
branches,  symmetrically  placed  with  respect  to  the  line  FF'. 


182 


ANALYTIC    GEOMETRY. 


167.    The  point  0,  half  way  between  the  foci,  is  the  Centre. 

The  line  AA'  passing  through  the  foci  and  limited  by  the 
curve  is  the  Transverse  Axis. 

The  points  A,  A',  Avhere  the  transverse  axis  meets  the 
curve,  are  called  the  Vertices. 

The  transverse  axis  is  equal  to  the  constant  difference  2a, 
and  is  bisected  by  the  centre  (§  IGG). 


Fig.  68. 

The  line  £B'  passing  through  0  perpendicular  to  AA' 
does  not  meet  the  curve  (§  166)  ;  but  if  B,  B'  are  two  points 
whose  distances  from  the  two  vertices  A,  A'  are  each  equal  to 
c,  then  BB'  is  called  the  Conjugate  Axis,  and  is  denoted  hf2b. 

Since  A  AOB  =  A  AOB',  OB  =  OB'=^b;  that  is,  the' con- 
jugate axis  is  bisected  by  the  centre. 

In  the  triangle  AOB,  OA  =  a,  OB  =  b,  AB  =  c;  hence 

c'  =  a'+b\ 

The  chord  passing  through  either  focus  perpendicular  to 
the  transverse  axis  is  called  the  Latus  Eectum,  or  Parameter. 

Note.  Since  a  and  h  are  equal  to  the  legs  of  a  right  triangle,  a 
may  be  either  greater  or  less  than  6;  hence  the  terms  ''major''  and 
''minor''  are  not  appropriate  in  the  hyperbola. 


THE    HYPERBOLA.  183 

168.  To  find  the  equation  of  the  hyperbola,  having  given  the 
foci,  and  the  constant  difference  2a. 

By  proceeding  as  in  the  case  of  the  ellipse  (§  131),  and  sub- 
stituting h-  for  c-  —  a^,  we  obtain 

g-|!  =  l-  [40] 

The  lengths  r,  r'  of  the  focal  radii  for  any  point  (x,  y)  are 
r  =  ex  —  a   and   r'  =  ex-j-  a. 

The  equations  of  the  ellipse  and  the  hyperbola  differ  only 
in  the  sign  of  b-.  The  equation  of  the  hyperbola  is  obtained 
from  that  of  the  ellipse  by  changing  +^^  to  —b-.     In  general 

Any  formula  deduced  from  the  equation  of  the  ellipse  is 
changed  to  the  corresponding  formula  for  the  hyperbola  by 
merely  changing  -{-V  to  — 5^,  or  b  to  5 V— 1. 

169.  A  discussion  of  equation  [40]  leads  to  the  following 
conclusions  : 

(i.)  The  curve  cuts  the  axis  of  x  at  the  two  real  points 
(a,  0)  and  (-a,  0). 

(ii.)  The  curve  cuts  the  axis  of  y  at  the  two  imaginary 
points  (0,  6V^)  and  (0,  -Z>V^). 

(iii.)  Xopart  of  the  curve  lies  between  the  straight  lines 
a:  =  -f  a  and  a;  =  —  a. 

(iv.)  Outside  these  lines  the  curve  extends  without  limit 
both  to  the  right  and  to  the  left. 

(v.)  The  greater  the  abscissa,  the  greater  the  ordinate. 

(vi.)  The  curve  is  symmetrical  with  respect  to  the  axis  of  x. 

(vii.)  The  curve  is  symmetrical  with  respect  to  the  axis  of  y. 

(viii.)  Every  chord  which  passes  through  the  centre  is  bi- 
sected by  the  centre.  This  explains  why  the  point  half  way 
between  the  foci  is  called  the  centre. 

The  two  distinct  parts  of  the  curve  are  called  the  right-hand 
and  the  left-hand  branches. 


184 


ANALYTIC    GEOMETRY. 


170.    An  hyperbola  who.-o   transverse   and   conjugate   axes 
are  equal  is  called  an  Equilateral  Hyperbola.     Its  equation  is 

cc-  — 2/-  =  «"^.  [41] 

The  equilateral  hyperbola  bears  to  the  general  hyperbola 
the  same  relation  that  the  auxiliary  circle  bears  to  the  ellipse. 


Fig.  69. 


i 


171.  The  hyperbola  which  has  BB'  for  transverse  axis,  and 
A  A'  for  conjugate  axis,  obviously  holds  the  same  relation  to 
the  axis  of  y  that  the  hyperbola  which  has  AA'  for  transverse 
axis  and  J3£'  for  conjugate  axis  holds  to  the  axis  of  .i\ 

Therefore  its  equation  is  found  by  simply  changing  the 
signs  of  a'  and  P  in  [40],  and  is 


=  1,    or 


=  -1. 


The  two  hyperbolas  are  said  to  be  Conjugate. 


THE    IIYrEIlBOLA.  185 

172.  The  straight  line  y  =  tyix,  passing  through  the  centre 
of  the  hyperbola  —^ — '77^  =  !,  meets  the  curve  in  two  points, 
the  abscissas  of  which  are 

-{-ah  —  ah 


Hence   the  points  will   be   real,  imaginary ,  or  situated  at 
infinity,  as  h'-  —  ni^a'^  is  positive,  negative,  or  zero  ;  that  is,  as 

in  is  less  than,  greater  than,  or  equal  to  -• 

The  same  line,  y  =  mx,  will  meet  the  conjugate  hyperbola 

-^  —  ^2  ~  ~  -^  i^  ^^^'^  points,  whose  abscissas  are 

—        +  ah  _        —  ah 


^rd^d^  —  h'^  ^m^d^  —  h'^ 

Hence  these  points  will  be  imaginary,  real,  or  situated  at 

infinity,  as  mi  is  less  than,  greater  than,  or  equal  to  — 

a 

Whence 

If  a  straight  line  through  the  centre  meet  an  hyperbola  in 
imaginary  points,  it  ivill  meet  the  conjugate  hyperbola  in  real 
points,  and  vice  versa. 

173.    An  Asymptote  is  a  straight  line  which  passes  through 
finite  points,  and  meets  a  curve  in  two  points  at  infinity. 
We  see  from  §  172  that  the  hyperbola 

a'      b' 
has  two  real  asymptotes  passing  through  the  centre  of  the 

curve,  and  having  for  their  equations  y~-{--x  and  y~ x  ; 

or,  expressed  in  one  equation, 

052         y2 


b' 


O.  [42] 


186  ANALYTIC    GEOMETRY. 

Ex.  36. 

What  is  the  equation  of  an  hyperbola,  if 

1.  Transverse  axis  =  16,  conjugate  axis  =  14  ? 

2.  Conjugate  axis  =  li2,  distance  between  foci  =  13  ? 
0.    Distance  between  foci  =  twice  the  transverse  axis  ? 

4.  Transverse  axis  =  8,  one  point  =  (10,  25)  ? 

5.  Distance  between  foci  =^  2(?,  eccentricity  =  V2  ? 

6.  Prove  that  the  latus  rectum  of  an  hyperbola  is  equal  to 
2^ 

7.  The  equation  of  an  hyperbola  is  9^:^  —  16?/^  =  25  ;  find 
the  axes,  distance  between  the  foci,  eccentricity,  and  latus 
rectum. 

8.  AVrite  the  equation  of  the  hyperbola  conjugate  to  the 
hyperbola  9x^  —  16y^  =  25,  and  find  its  axes,  distance  between 
its  foci,  and  its  latus  rectum. 

9.  If  the  vertex  of  an  hyperbola  bisects  the  distance  from 
the  centre  to  the  focus,  find  the  ratio  of  its  axes. 


10.  Prove  that  the  point  (x,  y)  is  without,  on,  or  within  the 

hyperbola,  according  as  —^—-.^  —  1  is  negative,  zero,  or  i^osi- 
tive. 

11.  Find  the  eccentricity  of  an  equilateral  hyperbola. 

12.  The  distance  of  any  point  of  an  equilateral  hyperbola 
from  the  centre  is  a  mean  proportional  between  its  focal  radii. 

13.  The  asymptotes  of  an  hyperbola  are  the  diagonals  of 
the  rectangle  CD FF  (Fig.  69,  p.  184). 

14.  Find   the   foci   and   the   asymptotes   of  the  hyperbola 
162;^-9y^  =  144. 


THE    HYPERBOLA.  187 

15.  The  asymptotes  of  an  equilateral  hyperbola  are  perpen- 
dicular to  each  other.  Hence  the  equilateral  hyperbola  is 
also  called  the  rectangular  hyperbola. 

16.  An  hyperbola  and  its  conjugate  have  the  same  asymp- 
totes. 

17.  Find  the  length  of  a  perpendicular  dropped  from  the 
focus  to  an  asymptote. 


Tangents  and  Normals. 

Note.  The  results  stated  in  the  following  six  sections  are  established 
in  the  same  way  as  the  corresponding  propositions  relating  to  tangents 
and  normals  to'afi  ellipse.     We  shall,  therefore,  omit  the  proofs. 

174.  The  equation  of  the  tangent  at  {x^,  y^  is 

^-^  =  1.  [43] 

175.  The  equation  of  the  normal  at  (xi,  iji)  is 

y^x   ,   oc,  1/      x,y,  (rt2  +  &2) 

~TT  "I ~  = TT-y •  V^^\ 

b'         a  a-b-  ^     ^ 

176.  The  subtan<Tent  = ^,  the  subnormal  =  — 5-^- 


177.  The  tangent  and  the  normal  at  any  point  of  an  hyper- 
bola bisect  the  angles  formed  by  the  focal  radii  of  the  point 
(>  144). 


178.  The  straight  line  whose  equation  is  y = mx  -. 
is  a  tangent  for  all  values  of  m  (§  146). 

179.  The  equation  of  the  director  circle  of  an  hyperbola  is 
x^j^y'^  =  a'-  h'  (§  147). 


188  ANALYTIC    GEOMETRY. 


Ex.  37. 


1.  Find  the  equations  of  tangent  and  normal  to  the  hyper- 
bola 16  x"^  —  9 y'^  —  112  at  the  point  of  contact  (4,4).  Also 
find  the  lengths  of  the  subtangent  and  the  subnormal. 

2.  Show  that  in  an  equilateral  hyperbola  the  subnormal  is 
equal  to  the  abscissa  of  the  point  of  contact. 

3.  The  equations  of  the  tangent  and  the  normal  at  a  point 
of  an  equilateral  hyperbola  are  5:r  — 4y  =  9,  4.r -f  9?/ =  56. 
What  is  the  equation  of  the  hyperbola,  and  what  are  the 
co-ordinates  of  the  point  of  contact  ? 

4.  For  what  points  of  an  hyperbola  is  the  subtangent  equal 
to  the  subnormal  ? 

5.  To  draw  a  tangent  and  a  normal  to  an  hyperbola  at  a 
given  point  of  the  curve. 

6.  If  an  ellipse  and  an  hyperbola  have  the  same  foci,  prove 
that  the  tangents  to  the  two  curves  drawn  at  their  points  of 
intersection  are  perpendicular  to  each  other. 

7.  Prove  that  the  asymptotes  of  an  hyperbola  are  tangent 
to  it  at  infinity. 

8.  Prove  that  the  length  of  a  normal  in  an  equilateral 
hyperbola  is  equal  to  the  distance  of  the  j)oint  of  contact  from 
the  centre. 

9.  Find  the  distance  from  the  origin  to  the  tangent  through 
the  end  of  the  latus  rectum  of  the  equilateral  hyperbola 
x^  —  'if'  =  a^. 

10.  What   condition  must  be   satisfied  in   order  that   the 

X  II  ?/'"         ?/'" 

straight  line h  -  =  1  niay  touch  the  hvperbola  ^  —  Vr,  =1? 

^  m      n  ^  '  ^  a       b' 

11.  When  is  the  director  circle  of  an  hyperbola  imaginary  ? 

12.  Find  the  locus  of  the  foot  of  a  perpendicular  dropped 
from  the  focus  of  an  hyperbola  to  a  tangent. 


THE    HYPERBOLA.  189 

Ex.  38.      (Review.) 

1.  The  ordinate  through  the  focus  of  an  hyperbola,  pro- 
duced, cuts  the  asymptotes  in  P  and  Q.  Find  PQ  and  the 
distances  of  P  and  Q  from  the  centre. 

2.  In  the  hyperbola  9:c^  —  IGy^  =  144  what  are  the  focal 
radii  of  the  points  whose  common  abscissa  is  8  ?  What  other 
points  have  equal  focal  radii  ? 

3.  What  relation  exists  between  the  surti  of  the  focal  radii 
of  a  point  of  an  hyperbola  and  the  abscissa  of  the  point  ? 

4.  Prove  that  in  the  equilateral  hyperbola  every  ordinate 
is  a  mean  proportional  between  the  distances  of  its  foot  from 
the  vertices  of  the  curve.  Hence  find  a  method  of  construct- 
ing an  equilateral  hyperbola  when  the  axes  are  given. 

5.  In  the  equilateral  hyperbola  the  distance  of  a  point  from 
the  centre  is  a  mean  proportional  between  its  focal  radii. 

6.  In  the  equilateral  hyperbola  the  bisectors  of  the  angles 
formed  by  lines  draw^n  from  the  vertices  to  any  point  of  the 
curve  are  parallel  to  the  asymptotes. 

7.  If  e,  e'  are  the  eccentricities  of  two  conjugate  hyperbolas, 

8.  Through  the  positive  vertex  of  an  hyperbola  a  tangent  is 
drawn.     In  what  points  does  it  cut  the  conjugate  hyperbola  ? 

9.  The  sum  of  the  reciprocals  of  two  focal  chords  perpen- 
dicular to  each  other  is  constant. 

10.  Through  the  foot  of  the  ordinate  of  a  point  in  an  equi- 
lateral hyperbola  a  tangent  is  drawn  to  the  circle  described 
upon  the  transverse  axis  as  diameter.  What  relation  exists 
between  the  lengths  of  this  tangent  and  the  ordinate  of  the 
point  ? 


190  ANALYTIC    GEOMETRY. 

11.  In  an  equilateral  hyperbola  find  the  equations  of  tan- 
gents drawn  from  the  positive  end  of  the  conjugate  axis. 

12.  From  what  point  in  the  conjugate  axis  of  an  hyperbola 
must  tangents  be  drawn  in  order  that  they  may  be  perpen- 
dicular to  each  other? 

13.  What  condition  must  be  satisfied  that  a  square  may  be 
constructed  whose  sides  shall  be  parallel  to  the  axes  of  an 
hyperbola  and  whose  vertices  shall  lie  in  the  curve? 

14.  Find  the  equation  of  the  chord  of  the  hyperbola 
16a:'  — 9?/' =  144  which  is  bisected  at  the  point  (12,3). 

15.  Find  the  equation  of  a  tangent  to  the  hyperbola 
16a;'  — 9?/' =  144  parallel  to  the  line  ?/  =  4.r  — 3. 

16.  Determine  the  points  in  an  hyperbola  for  ^vhich  the 
length  of  the  tangent  is  twice  that  of  the  normal. 

17.  A  chord  of  an  hyperbola  which  touches  the  conjugate 
hyperbola  is  bisected  at  the  point  of  contact. 

SUPPLEMENTARY   PROPOSITIONS. 

Note.  Many  of  the  following  propositions  are  closely  analogous  to 
propositions  already  established  for  the  ellipse  ;  hence  the  proofs  are 
omitted,  and  references  given  to  the  chapter  on  the  ellipse. 

180.    Two  tangents  can  he  drawn  to  an  hyperbola  from  any 

point  (A,  h) ;  and  they  will  be  reed,  coincident,  or  imaginary, 

as  the  point  is  without,  on,  or  within  the  curve  (^  148). 

A'      B 

The  two  tancrents  will  be  real  if  — -  —  1  is  negative. 

a'      y^ 

Likew^ise  two  real  tangents  can  be  drawn  from  (A,  Ic)  to  the 
conjugate  hyperbola  if  — 2~72"^-^  ^^  negative. 


THE    HYPERBOLA.  191 

Hence  it  follows  that  if  —  —    -,  lias  any  value    between    0 
a-      0^ 

and  —  2,  a  pair  of  real  tangents  can  be  drawn  from  (A,  Ic)  to 
each  hyperbola, 

181.  The  straight  line  passing  through  the  points  of  contact 
of  the  two  tangents  drawn  to  an  hyperbola  from  any  point  P 
is  called  the  Polar  of  P  with  respect  to  the  hyperbola  ;  and  P 
is  called  the  Pole  of  this  straight  line. 

The  polars  of  the  foci  are  called  the  Directrices. 
The  equation  of  the  polar  of  the  point  (h,  k)  is 

^-^  =  1.  (§149) 

The  propositions  in  §§  80-82  hold  true  for  poles  and  polars 
with  respect  to  an  hyperbola,  and  may  be  proved  in  the  same 
way. 

182.  The  locus  of  the  middle  points  of  chords  parallel  to 

the  line  ?/  =  7?ix  is  72 

l/  =  ~  (§152) 

a'm 

This  locus  is  called  a  Diameter  of  the  hyperbola. 
Every  diameter  passes  through  the  centre. 
The  chords  bisected  by  a  diameter  are  called  the  Ordinates 
of  the  diameter. 

183.  If  m'  is  the  slope  of  the  diameter,  bisecting  chords 
parallel  to  the  line  y  =  mx,  then 

in?n=^J  [45] 

and  from  the  symmetry  of  this  equation  we  infer  that 

If  one  diameter  bisects  all  chords  parcdlel  to  another,  the 
second  diameter  will  bisect  all  chords  parallel  to  the  first. 

Two  diameters  drawn  so  that  each  bisects  all  chords  par- 
allel to  the  other  are  called  Conjugate  Diameters  (§  153). 


192  ANALYTIC    GEOMETRY. 

184.  Since  the  product  of  the  slopes  of  two  conjugate  diam- 
eters is  ip- 

it  follows  that  vi  and  m'  must  agree  in  sign  ;  therefore 
Two  conjugate  diameters  lie  in  the  savie  quadrant. 

Also,  if  m  in  absolute  magnitude  is  less  than  -,  then  m' 

h  ^^ 

must  be  greater  than  —     But  the  slope  of  the  asymptotes  is 

h  ^^ 

equal  to  ±  -     Therefore 
a 

Two  conjugate  diameters  lie  on  ojyposite  sides  of  the  asymp- 
tote in  the  same  quadrant ;  and  of  two  conjugate  diam,eters,  one 
Tneets  the  curve  in  real  points  and  the  other  in  imaginary  points 
(§  172). 

185.  The  length  of  a  diameter  which  meets  the  hyperbola 
in  real  points  is  the  length  of  the  chord  between  these  points. 

If  a  diameter  meets  the  hyperbola  in  imaginary  points, 
that  is,  does  not  meet  it  at  all,  it  will  meet  the  conjugate 
hyperbola  in  real  points  (§  172)  ;  and  its  length  is  the  length 
of  the  chord  between  these  points. 

A  comparison  of  the  equations  of  two  conjugate  hyper- 
bolas will  show  that  if  a  diameter  meet  one  of  the  hyper- 
bolas in  the  imaginary  point  (AV—  1,  Z*V—  1),  it  will  meet 
the  other  in  the  real  point  (A,  h)  ;  hence  the  length  of  the 
semi-diameter  will  be  VA^  -f-  U\ 

186.  The  equations  of  an  hyperbola  and  its  conjugate  differ 
only  in  the  signs  of  a?-  and  h^.  But  this  interchange  of  signs 
does  not  effect  the  equation 

m??i'  =  — .     Therefore 
d^ 

If  two  diameters  are  conjugate  with  respect  to  one  of  two  con- 

jugate  hxjperholas,  they  loill  he  conjugate  with  respect  to  the  other. 


THE    IIYrERBOLA. 


193 


Thus,  let  POP  and  QOQ'  (Fig.  70)  be  two  conjugate  di- 
ameters. Then  FOP  bisects  all  chords  parallel  to  QOQ'  that 
lie  luiihin  the  branches  of  the  original  hyperbola  and  between 
the  branches  of  the  conjugate  hyperbola  ;  and  QOQ'  bisects  all 
chords  parallel  to  POP'  that  lie  within  the  branches  of  the 
conjugate  hyperbola  and  between  the  branches  of  the  original 
hyperbola. 

% 

1C/ 


By 


'B 


M 


P? 


[B' 


Fig.   70. 


From  the  above  theorem  it  follows  immediately  that 

If  a  straight  line  nncet  each  of  two  conjugate  hyperbolas  in 
two  real  points,  the  two  portions  of  the  line  contained  between 
the  liyperbolas  are  equal  {thus,  BD  =  B'D',  Fig.  70). 

187.  TJie  tangent  drawn  through  the  end  of  a  diameter  is 
parallel  to  the  conjugate  diameter  (§  154). 


194  ANALYTIC    GEOMETRY. 

188.    Having  given  tlie  end  {xi,  y{)  of  a  diameter,  to  find  the 
end  {x.,  1/2)  of  ike  conjugate  diameter. 

If  (.rj,  ?/i)  is  a  point  of  the  hyperbola 

1-f!=i.  (1) 

a^      0 

we  know  (§  172)  that  (5:2, 3/2)  will  be  a  point  of  the  conjugate 

hyperbola  2      ,,2 

--•^  =  -1.  (2) 

If  the  equation  of  the  diameter  through  {x^,  3/1)  be 
y  =  mx, 

then  m  =  — 

Xi 

If  the  equation  of  the  diameter  through  (x^,  y^  be 
y  ==  m^x, 

then  nn^  =  -^   (§  182). 

am 

Hence  the  equation    y  =  7n'x  may  be  written 

y-^-  (3) 

The  values  of  x^  and  3/2  are  now  found  by  solving  equations 
(2)  and  (3),  and  are 

a  h 

-2/1,     y-i  =  ±  -^*i. 


b^ 


a 


The  positive  signs  belong  to  one  end,  and  the  negative  signs 
to  the  other  end,  of  the  conjugate  diameter. 

189.    If'  0  deyiote   the  angle  formed  by  two  conjugate  scini- 
diameters,  a'  and  b',  their  lengths,  then  sin  6  =  — —  (§  15S). 


THE    IIYPEEBOLA.  195 

190.  To  fuid  the  equation  of  an  hyperbola  referred  to  any 
pair  of  conjugate  diameters  as  axes  of  eo-ordinates. 

If  a',  b'  denote  the  two  semi-diameters,  the  required  equa- 
tion is                                 ,2         -i 

^ ^  =1.  (1) 

a"      b''  ^^ 

The  method  of  solving  the  problem  is  the  same  as  that  used 
in  §  IGO  ;  but  since  the  intercept  of  the  curve  on  the  axis  of  y 
is  imaginary,  §  167,  the  sign  of  ^  in  §  160  will  be  negative. 

Since  the  form  of  equation  (1)  is  the  same  as  that  of  the 
equation  referred  to  the  axes  of  the  curve,  it  follows  that  all 
formulas  which  have  been  obtained  without  assuming  the  axes 
of  co-ordinates  to  be  at  right  angles  to  each  other  hold  good 
when  the  axes  of  co-ordinates  are  any  two  conjugate  diam- 
eters. For  example,  the  equation  of  the  asymptotes  of  the 
hyperbola  re^Dresented  by  equation  (1)  is 

^--^--=0. 
a''      ¥' 

191.  The  tangents  through  the  ends  of  two  conjugate  diam- 
eters meet  in  the  asymptotes. 

Let  the   ends  of  the  diameters  be  the  points  {x^,  y^  and 

fe,  2/2)  ;  i^Q"^  the  equations  of  tangents  through  {x^,  y^  and 

(^2, 2/2)  will  be              x^  _y{y__  .,^ 

a'  ~     b'         '  ^  ^ 

and  ^_^=:_1.  (2) 

a         b^  ^  ^ 

The  asymptote  y^=-'x  meets  the  tangent  represented  by 
(1)  in  the  point  /      a:b  ah" 


})x^—  ay^^bx^  —  ay,^ 
and  the  tangent  represented  by  (2)  in  the  point 
/      ccb  ab"- 

\ay2  —  bx2     ay-i  —  bx.^, 


196  ANALYTIC    GEOMETRY. 

But  from  §  188,      hx^  =  ay^   and    aij2  =  bxi. 

Therefore  ay2  —  hx.j,  =  hx^  —  ayy. 

Hence  we  see  that  the  two  jDoints  of  meeting  coincide. 

192,  To  find  under  what  condition  an  equation  will  repre- 
sent an  hyperbola  ivhen  of  the  form 

Ax'  +  ^y  +  Dx  -\-Ey+F=0. 

If  neither  A  nor  B  is  zero,  the  equation  may  be  w^ritten 

By  proceeding  as  in  §  161,  we  find  that  the  equation  repre- 
sents in  general  an  hyperbola  if  A  and  B  are  neither  of  them 
zero,  and  have  unlike  signs. 

The  axes  of  the  hyperbola  are  parallel  to  the  axes  of  x  and  y, 

f     T)        y 

and  the  centre  is  the  point  f 7' - 

193.  If  a  straight  line  cut  an  hyperbola  and  its  asymptotes 
the  p)ortions  of  the  line  intercepted  between  the  curve  and  its 
asymptotes  are  equal. 

Let  CC^  (Fig.  71)  be  the  line  meeting  the  asymptotes  in  C,  C" 
and  the  curve  in  B,  B\  and  let  the  equation  of  the  line  be 

y  =  ')nx-\-c.  (1) 

Let  il/be  the  middle  point  of  the  chord  BB' ;  then  (§  182) 
the  equation  of  the  diameter  through  m  is 

y  =  ^-  (2) 

am 

By  combining  equation  (1)  with  the  equations  of  the  asymp- 
totes, we  obtain  the  co-ordinates  of  the  points  (7  and  C"  ;  taking 


THE    IIYPERLOLA. 


197 


the  half-sum  of  these  values,  we  get  for  the  co-ordinates  of  the 
point  halfway  between  (7  and  C"  the  values 


Dj 


M 


X 


P? 


KB' 


Q' 


Fig.  71. 


These  values  satisfy  equation  (2)  ;  therefore  the  point  half 
way  betw^een  Cand  C  coincides  with  M;  therefore  MC^MC. 
And  since  3IB  =  MB\  therefore  BC=B'C\ 

Let  CC^  be  moved  parallel  to  itself  till  it  becomes  a  tangent 
at  P,  meeting  the  asymptotes  m  R,  8\  then  the  points  B,  B' 
coincide  at  P,  and  we  have  PB  =  P8.     Hence 

The  portion  of  a  tangent  intercepted  by  the  asymptotes  is 
bisected  by  the  point  of  contact. 


198 


ANALYTIC    GEOMETRY. 


194.    To  find  the  equation  of  an  lujijerhola  referred  to  the 
asymptotes  as  axes  of  co-ordinates. 


Fig.   72. 


Let  the  lines  OB,  OC  (Fig.  72)  be  the  asymptotes,  A  the 
vertex  of  the  curve,  and  let  the  angle  AOC='  a. 

Let  the  co-ordinates  of  any  point  P  of  the  curve  be  x,  y 
when  referred  to  the  axes  of  the  curve,  and  x\  y'  when  referred 
to  OB,  OC  9.S  axes  of  co-ordinates. 

Draw  F3I±  to  OA,  FNW  to  OC;  then 

x  =  03f,     y  =  MP,  x'^ON,     y'  =  NP. 
x  =  ON  cos  a  -j-  NP  cos  a  =  (.r'  +  y')  <^os  a, 
y  =  NP  sin  a  —  ON  sin  a  =  (y'  —  x^)  sin  a. 

Hence,  by  substitution  [40],  we  obtain 

{x'  -f  y^y  cos'^ a      (?/'  —  a-')-  sin^ a 


But  from  the  relation  tan  a 


1. 


we  have 


sm  a 


cos*  a 


a'  +  ¥  a'  -f  b' 

Substituting  these  values,  and  dropping  accents,  we  have 

4x?/ =  a- +  6-.  [46] 


THE    HYPERBOLA.  199 

195.  The  following  method  of  showing  that  an  hyperbola 
has  asymptotes,  and  finding  their  equations,  is  more  general 
than  the  method  given  in  §§  172,  173. 

The  abscissas  of  the  points  where  the  straight  line  y  =  mx-\-c 
meets  an  hyperbola  are  found  by  solving  the  equation 

a;"      {mx  +  <^)^  _  i 


Ir  —  7?i^a^   2      2  inc  1/  +  c^ 

x^ : — X—        ' 


(1) 


ieb'  b'  b' 

This  equation  is  identical  with  the  equation 

Ax'  +  2Bx-^C=0  (2) 

c  A       b^'  —  ni^a?       T)  r>ic      ^  b"^ '\- c^ 

it  Ji  =  ; >         1j   — — ->  O  = ; 

d'h'  b'  b' 

If  Xi,  x-i  are  the  roots  of  (2), 


^  _-B  +  ^B'-AC C 

^  -B--^~B'-AC 

-B~-^B'-AC C_ 

^  -B  +  VB'-AC 


c 

If  ^  =  0  and  also  ^  =  0,  we  have  ^^i  =  .Tg  =  —  =  co  ;  and 

the  line  y  ^^  mx -{- c  w^ll  meet  the  curve   in   two  points  at 
infinity. 

1(  A  =  0,  m  —  zt  —     li  B  =  0,  c  =  0.     Therefore  there  are 

,  "            ■                      *          ,              b 
two  asymptotes  ;  their  equations  are  ?/  =  ~x  and  y  = x. 

If  only  A  =  0,  then  ni  =  rb  -<  the  straight  line  is  2'>arallel 

to  an  asymptote,  and 

ar,  =  —  —  =■■ ^^^—  J     X2^=  oo.       Therefore 

2B  2mc 

A  straight  line  parallel  to  an  asymptote  meets  the  curve  in 
one  finite  point  and  in  one  point  at  infinity. 


200 


ANALYTIC    GEOMETRY. 


196.  To  find  the  locus  of  a  point  which  moves  so  that  its 
distance  from  a  fixed  point  bears  to  its  distance  from  a  fixed 
straight  line  a  constant  ratio  greater  than  unity. 


Fig.  73. 


Let  e  =  the  constant  ratio,  and  2p  =  the  distance  from  the 

fixed  point  F  (Fig.  73)  to  the  fixed  line  DN.     Choosing  for 

axes  the  line  drawn  through  i^  perpendicular  to  DN,  and  the 

fixed  line  DN,  we  obtain  the  same  equation  as  that  found  in 

§  162, 

^         '  (l-eV  +  y2-4;?:r+4/-0.  (1) 

But  since  e  is  greater  than  1,  the  equation  now  will  repre- 
sent an  hyperbola  (§   195),   the  centre   of  which  is  at  the 

2p 


(2) 


point  [  T    "  J  0 

Transferring  the  origin  to  the  centre  0,  we  get 


2ep)      ^{2epy 


.r 


1. 


THE    IIYl'ERBOLA.  201 


Putting  a  =  -,^-^2'  h  =    ^^?--  (2)  becomes 
-'-      ^  we  — 1 

the  equation  of  an  hyperbola  in  the  ordinary  form. 

It  may  be  shown,  as  in  §  162,  that  the  fixed  point  F  coin- 
cides with  the  focus  of  the  curve,  and  that  the  constant  ratio 
e  is  equal  to  the  eccentricity. 

This  locus  is  often  taken  as  the  definition  of  an  hyperbola, 
F  being  called  the  Focus,  and  the  fixed  line  DN  the  Directrix. 

The  symmetry  of  the  curve  with  respect  to  the  conjugate 
axis  shows  that  there  are  two  foci  and  two  directrices  sym- 
metrically placed  with  respect  to  the  conjugate  axis. 

197.  To  find  the  polar  equation  of  an  hyperhola,  the  left- 
hand  focus  being  taken  as  pole. 

For  a  point  in  the  right-hand  branch,  the  distance  to  the 
pole,  which  is  the  remote  focus,  is  (§  168) 

p  =  ex  -}-  a, 

the  distance  x  being  reckoned  from  the  centre. 

Now  X  =  p  cos  6  —  c  =  p  cos  0  —  ae. 

Whence,  by  substitution  and  reduction. 


e  cos  0  —  1 


[47] 


Ex.  39. 

1.  "What  is  the  polar  of  the  point  (—  9,  7)  with  respect  to 
the  hyperbola  7^*  -  12?/^  =  112  ? 

2.  Find  the  equations  of  the  directrices  of  an  hyperbola. 

3.  Find  the  angle  formed  by  a  focal  chord  and  the  line 
which  joins  its  pole  to  the  focus. 


^\'^2         Vxlh n. 


202  ANALYTIC    GEOMETRY. 

4.  Find  the  pole  of  the  line  Ax-\-By-\-  C  with  respect  to 
an  hyperbola. 

5.  Find  the  polar  of  the  right-hand  vertex  of  an  hyperbola 
with  respect  to  the  conjugate  hyperbola. 

6.  Find  the  distance  from  the  centre  of  an  hyperbola  to 
the  point  where  the  directrix  cuts  the  asymptote. 

7.  If  (.Tx,  ?/i)  and  (^^2, 1/2)  are  the  ends  of  two  conjugate 
diameters,  then 

8.  The  equation  of  a  diameter  in  the  hyperbola  25a'^— 16y^ 
=  400  is  Zy  =  x.  Find  the  equation  of  the  conjugate  diam- 
eter. 

9.  In  the  hyperbola  49 rr^  —  4y-  =  196,  find  the  equation  of 
that  chord  which  is  bisected  at  the  point  (5,  3). 

10.  Find  the  length  of  the  semi-diameter  conjugate  to  the 
diameter  y  =  3.r  in  the  hyperbola  ^x^  —  4?/^  =  36. 

11.  The  area  of  the  parallelogram  formed  by  drawing  tan- 
gents through  the  ends  of  two  conjugate  diameters  is  constant, 
and  equal  to  4aZ>. 

12.  If  a\  h'  are  the  lengths  of  two  conjugate  semi-diameters, 
then  ^,r2_5r2^^2_  j2 

13.  Prove  that  PQ  (Fig.  70)  is  parallel  to  one  asymptote 
and  bisected  by  the  other. 

14.  An  asymptote  is  its  own  conjugate  diameter. 

15.  The  conjugate  diameters  of  an  equilateral  hyperbola 
are  equal. 

16.  Having  given  two  conjugate  diameters  in  length  and 
position,  to  find  by  construction  the  asymptotes  and  the  axes. 

17.  To  draw  a  tangent  to  an  hyperbola  from  a  given  point. 


THE    HYPERBOLA.  203 

18.  Find  the  equation  of  a  conjugate  hyperbola  referred  to 
its  asymptotes  as  axes. 

19.  Find  the  equation  of  a  tangent  at  any  point  (x^,  3/1)  of 
the  hyperbola  4  .ry  =  a^  +  ^^ 

20.  Find  the  equation  of  an  hyperbola,  taking  as  the  axis 

^       (i.)    the  tangent  through  the  left-hand  vertex  ; 
(ii.)  the  tangent  through  the  right-hand  vertex. 

21.  Trace  the  form  of  an  hyperbola  by  means  of  the  polar 
equation,  p.  201. 

22.  Find  the  polar  equation  of  an  hyperbola,  taking  the 
right-hand  focus  as  pole. 

23.  Find  the  polar  equation  of  an  hyperbola,  taking  the  ■ 
centre  as  pole. 

24.  Show  that  the  equation 

a:--?/'-2p;-4y  +  l  =  0 
represents  an  hyperbola.     Find  its  centre  and  axes,  and  con- 
struct roughly  the  curve. 

25.  The  distance  from  a  fixed  point  to  a  fixed  straight  line 
is  10.  Find  the  locus  of  a  point  which  moves  so  that  its  dis- 
tance from  the  fixed  point  is  always  twice  its  distance  from 
the  fixed  line. 

26.  Through  the  point  (—  4,  7)  a  straight  line  is  drawn  to 
meet  the  axes  of  co-ordinates,  and  then  revolved  about  this 
point.     Find  the  locus  of  its  middle  point. 

27.  A  straight  line  has  its  ends  in  two  fixed  perpendicular 
lines,  and  forms  with  them  a  triangle  of  constant  area  «^ 
Find  the  locus  of  the  middle  point  of  the  line  (see  §  000). 

28.  The  base  a  of  a  triangle  is  fixed  in  length  and  position, 
and  the  vertex  so  moves  that  one  of  the  base  angles  is  always 
double  the  other.     Find  the  locus  of  the  vertex. 


CHAPTER   VIII. 
LOCI    OF    THE    SECOND    ORDER. 

198.  The  locus  represented  by  an  equation  of  the  second 
degree  is  called  a  Locus  of  the  Second  Order. 

We  have  seen,  in  the  preceding  chapters,  that  the  circle, 
parabola,  ellipse,  and  hyperbola  are  loci  of  the  second  order. 
We  now  propose  to  inquire  whether  there  are  other  loci  of 
the  second  order  besides  the  four  curves  just  named;  in  other 
words,  to  find  what  loci  may  be  represented  by  equations  of 
the  second  degree. 

For  this  purpose  we  shall  write  the  general  equation  of  the 
second  degree  in  the  form 

Ax'  +  ^/  +  Ox2j  +  Dx+i:i/-\-F=0,  (1) 

and  shall  assume  that  the  axes  of  co-ordinates  are  rectangular. 
This  assumption  will  in  nowise  diminish  the  generality  of 
our  conclusions  ;  for  if  the  axes  be  supposed  oblique,  we  can 
change  them  to  rectangular  axes,  and  this  change  will  not 
alter  the  degree  of  the  equation  or  the  nature  of  the  locus 
which  it  represents  (§  100). 

199.  If  we  suppose  the  coefficients  of  equation  (1)  to  be 
susceptible  of  all  values  including  zero,  we  see  that  (1)  in- 
cludes the  general  equation  of  the  first  degree  as  a  special 
case  when  A  =  B  =  C=  0. 

But  even  if  no  one  of  the  coefficients  is  zero,  they  may 
stand  in  such  a  relation  to  one  another  that  the  equation  can 
be  resolved  into  two  linear  factors,  and  therefore  represents 
straight  lines,  real  or  imaginary. 


LOCI    OF   THE    SECOND    ORDER.  205 

In  order  to  find  what  this  relation  is,  let  us  solve  (1)  with 
resj^ect  to  one  of  the  variables.  Choosing  y  for  this  purpose, 
we  obtain  n  ^  \p        \ 

y  =  --4^±2l^VSM^l^Ti^  (2) 

where 

L  =  C''-^AB,    M=1{CE--2BB),   N=E''-^BF. 

If  Lx^  A^Mx-\-N  be  the  square  of  a  binomial  of  the  form 
&x  +  T,  then  the  value  of  y  may  be  written 

Cx^E      Sx  +  T 

y  =  —  ■ — ■■  ± — . 

-^  2B  2B 

and  the  locus  of  (1)  will  in  general  be  a  pair  of  straight  lines. 
Xow,    from    Algebra,    we    know    that    the    condition    that 
Bx'^  +  Mx  +  iV  should  be  a  perfect  square  is 

]IP-4:BjV=0; 
or,  substituting  the  values  of  B,  M,  and  iV, 

(CE -  2BBy -  (C  -  ^AB)  (E'  -  4.BF)  =  0, 

or  F(C'~4:AB)+AE'+BB'-CBE=0.       (3) 

The  quantity  on  the  left-hand  side  of  equation  (3)  is  usually 
denoted  by  A,  and  is  called  the  Discriminant  of  equation  (1). 
And  we  may  conclude  that  (1)  represents  straight  lines  (real 
or  imaginary)  whenever  A  —  0. 

200.  In  order  to  simplify  the  form  of  equation  (1),  let  us 
change  the  origin  to  the  point  (A,  k),  and  then  so  choose  the 
values  of  h  and  k  that  the  terms  involving  the  first  powers  of 
X  and  y  will  vanish.  Making  the  change  by  substituting  in 
(1)  x-^  h  for  X,  and  y-f  k  for  ?/,  we  find  that  the  coefficients 
A,  B,  and  C  remain  unaltered,  and  we  may  write  the  trans- 
formed equation 

Ax'  -f  By'  4-  Cxy  +  B'x  i-E'y  =  B  (4) 


206  ANALYTIC    GEOMETRY. 

where  D' =  2^A -f- a  + A 

E  =-[Ah^  +  BU'-^Chh  +  Dh  +  Eh-\-F]. 

The  values  of  h  and  Ic  which  will  make  i>'  and  E  vanish 
are  evidently  found  by  solving  the  equations 

2Ah+Ck+n  =  0, 
2EIc-\-Ch  +  E  =  0, 


4AE-C"  4:A£-C' 

The  value  of  E  can  now  be  reduced  to  a  form  very  easily 
remembered  : 

E  =  -  [Ah'  +  EIc'  -f  Chk  +  Eh+  Ek  +  E] 

=  -  ^[{2.Ah+Ck+D)h-\-{2Bk-^Ch+E)k^nh^Ek+2E'] 
=  -  ^{D'h  •+  E'k  +  Eh  +  Ek  +  2E) 
■=-i(Eh  +  Ek  +  2E) 

^      ,2BE'  -  CEE+  2AE'-CEE+  2E(C'-4:AB) 
'  C'-4:AB 

2' 

where  ^  =  4:AB~C\ 

Equation  (4)  may  now  be  written 

Ax'+By'  +  Cx7/=E.  (5) 

From  the  form  of  (5)  we  see  that  if  (x,  y)  be  a  point  of  the 
locus,  so  also  is  {—x,  —y)  ;  that  is,  the  new  origin  is  a  point 
so  placed  that  it  bisects  every  chord  passing  through  it.  A 
point  having  this  property  is  called  a  Centre  of  a  locus  of 
the  second  order. 

The  values  of  h  and  k  are  evidently  single  ;  hence  a  locus 
of  the  second  order  cannot  have  more  than  one  centre. 

The  values  of  h  and  k  are  finite,  provided  2  or  4:AB  —  C 
is  not  zero.     If,  however,  ^  =  0,  the  values  of  h  and  k  become 


LOCI    OF   THE    SECOND    ORDER.  207 

infinite  or  indeterminate.  In  this  case  a  change  of  origin  to 
the  centre  is  obviously  impossible,  and  a  different  method  of 
reduction  must  be  found. 

Hence  it  will  be  convenient  to  divide  loci  of  the  second 
order  into  two  classes  :  those  which  have  a  finite  centre,  and 
those  which  do  not.  The  ellipse  and  the  hyperbola  belong 
to  the  first  class  ;   the  parabola,  to  the  second  class. 

The  class  to  which  the  locus  of  a  given  equation  belongs 
is  ascertained  by  seeing  whether  the  value  of  2,  namely, 
AAB  —  C'\  is  or  is  not  equal  to  zero;  on  this  account  2 
may  be  called  the  Criterion  of  the  general  equation  of  the 
second  degree. 

CLASS   I.     :S  NOT   ZERO. 

201.  Equation  (5)  is  the  general  equation  of  loci  of  the 
second  order  wdiich  have  a  centre,  referred  to  the  centre  as 
the  origin  of  co-ordinates. 

If  in  equation  (5)  we  place  a:  =  0,  we  obtain  two  values  of 
y  equal  in  magnitude  and  opposite  in  sign.  Since  the  axis  of 
1/  is  not  limited  as  to  direction,  we  infer  that  cveri/  chord  pass- 
ing through  the  centre  is  bisected  at  the  centre.  Hence  a  chord 
passing  through  the  centre  is  called  a  Diameter. 

We  can  get  rid  of  the  term  involving  xg  by  another  change 
of  axes.  For  this  purpose  we  must  change  the  direction  of 
the  axes  through  an  angle  6,  keeping  the  origin  unaltered, 
and  then  determine  the  value  of  $  by  putting  the  new  term 
which  involves  xg  equal  to  zero. 

The  change  is  made  by  substituting  for  x  and  g,  in  equation 
(5),  the  respective  values  (§  95), 

X  cosO  —  g  sin  6, 
5;  sin  0-}-g  cos  0  ; 

and  equation  (5)  now  becomes 

Fx'  +  Qg'  +  C'xg  =  B, 


208  ANALYTIC    GEOMETRY. 

where  F  =  A  co.s"  0  +.11  siir  O+CainO  cos  0,  (6) 

Q  =Asm'0+J3co^'0-Csme  cos  6,  (7) 

C'  =  2(B-A)sinecoiiO-i-C(coii'0-sm'0).     (8) 

Putting  C  =  0,  we  obtain 

(A  -B)  sh)^J  —  Cco^,e  =  0,  (9) 

or  tan  2^  =  -—^,  (10) 

A—B 

a  relation  which  will  always  give  real  values  for  6,  since  the 
tangent  of  an  angle  may  have  any  value,  positive  or  negative. 

Since  the  angles  which  correspond  to  a  given  value  of  a 
tangent  differ  by  180°,  the  values  of  6  obtained  from  (10)  will 
differ  by  90°  ;  hence  the  new  axes  determined  by  (10)  are 
limited  to  a  single  definite  pair  of  perpendicular  lines,  passing 
through  the  centre. 

The  coefficients  P,  Q,  A,  B,  C  are  connected  by  simple 
relations,  which  are  independent  of  the  value  of  6,  and  which 
may  be  found  as  follows. 

From  (6)  and  (7),  by  addition  and  subtraction, 

Bi-Q  =  A+B,  (11) 

F~Q  =  (A-B)  co^^O  +  Csirfe.  (12) 

Equation  (9)  may  be  written 

0  =  (A-B)  siTi\e  -  Cco^'^0.  (13) 

Adding  the  squares  of  (12)  and  (13),  we  have 

(P  _  Qf  =  {A-  By  +  C\  (14) 

F-Q    =±^{A^W+^''  (IS) 

Whence,  from  (11)  and  (15), 

F=^^[A  +  B±^{A-By  +  C'l  (16) 

Q  =  i[A+B=fV{A-By  +  C-^].  (17) 

These  values  of  F  and  Q  are  always  real. 


LOCI    OF    THE    SECOND    ORDER.  209 

Finally,  by  squaring  (11)  and  subtracting  (14),  we  obtain 

4:FQ  =  4:AIl-C'^l.  (18) 

In  applying  these  formulas  the  question  arises  which  sign 
should  be  chosen  before  the  radical  in  equation  (15).  If  we 
take  for  20  the  smallest  positive  angle  which  corresponds  to 
the  value  of  tan  2^  in  (10),  then  the  value  of  2^  must  lie 
between  0°  and  180°,  and  sin  20  must  be  positive.  If,  now, 
in  equation  (12)  we  substitute  for  cos  2^,  its  value,  obtained 
from  (9),  equation  (12)  will  become 

0 

The  form  of  this  equation  shows  that  JP—  Q  must  have  the 
savie  sign  as  that  of  C. 

Equation  (5)  is  now  reduced  to  the  simple  form 

Px'-\-Qu'=E.  (19) 

We  see  from  (19)  that  the  axes  of  co-ordinates  are  now 
so  placed  that  each  axis  bisects  all  chords  parallel  to  the 
other  axis.  Two  lines  so  drawn  through  the  centre  of  a  curve 
of  the  second  order  that  they  have  this  property  are  called 
Conjugate  Diameters. 

From  what  proceeds  we  may  infer  that  in  those  loci  of  the 
second  order  which  have  a  centre  there  exists  one,  and  only 
one,  pair  of  rectangular  conjugate  diameters. 

These  two  diameters  are  called  the  Axes  of  the  curve.  Hence 
equation  (19)  is  the  general  equation  of  curves  of  the  second 
order,  referred  to  the  centre  as  origin  and  to  their  axes  as  the 
axes  of  co-ordinates. 

202.  The  nature  of  the  locus  represented  by  equation  (19) 
depends  upon  the  signs  of  P,  Q,  and  R.  There  are  two 
groups  of  cases,  according  as  5  is  positive  or  negative,  and 
three  cases  in  each  group. 


210  ANALYTIC    GEOMETRY. 

Group  1.     2  Positive. 

Then  equation  (18)  shows  that  P  and  Q  must  agree  in  sign. 
{A  and  B  must  also  agree  in  sign.) 

Case  1.  li  R  agree  in  sign  with  P  and  Q,  then  (§  135) 
the  locus  is  an  ellipse,  having  for  semiaxes 

Case  2.  If  i?  differ  from  P  and  Q  in  sign,  no  real  values 
of  X  and  y  will  satisfy  (19),  so  that  no  real  locus  exists.  But 
it  is  usual  to  say  in  this  case  that  the  locus  is  an  imaginary 
ellipse. 

Case  3.  If  ^  =  0,  the  locus  is  a  single  point,  namely,  the 
origin. 

Group  2.     %  Negative. 

Then  (18)  shows  that  P  and  Q  must  have  unlike  signs. 

Case  1.  If  i?  agrees  in  sign  with  P,  we  may,  by  division 
(and  changing  the  signs  of  all  the  terms  if  necessary),  put 
equation  (19)  into  the  form  of  equation  [40],  page  183. 
Therefore  the  locus  is  an  hyperbola,  with  its  transverse  axis 
on  the  axis  of  x,  and  having  for  semiaxes 

Case  2.  If  R  agrees  in  sign  with  Q,  we  may,  by  division 
(and  change  of  signs  if  necessary),  put  equation  (19)  into  the 
form  of  equation  (2),  page  184.  Therefore  the  locus  is  an 
hyperbola,  with  its  transverse  axis  on  the  axis  of  y. 

Case  3.  If  P  =  0,  the  locus  consists  of  two  straight  lines, 
intersecting  at  the  origin,  and  having  for  their  equations 


y-^-'l^ 


X. 


LOCI   OF    THE   SECOND    ORDER.  211 


CLASS  II.     2  =  0. 


203.  Let  us  now  suppose  that,  in  the  general  equation,  2, 
or  4:AB  —  (7^  is  equal  to  zero.  When  this  is  the  case,  A  and 
B  must  have  like  signs,  and  we  shall  assume  that  A  and  B 
are  positive  ;  if  they  happen  to  be  negative,  we  may  make 
them  positive  by  multiplying  the  equation  by  the  factor  —1. 

The  existence  of  a  curve  of  this  class  is  immediately  shown 
by  the  form  of  the  equation  ;  for  the  condition  4:AB  —  C^  =  0 
is  also  the  condition  that  the  first  three  terms,  Ax'-'rBi/'^-^-Cxy, 
form  a  complete  square. 

Throughout  this  section  we  shall  also  assume  that  C  is 
not  zero  ;  this  being  assumed,  it  follows  from  the  relation 
^AB  —  (7^  =  0  that  neither  A  nor  B  can  be  zero. 

When  2  =  0,  the  values  of  the  co-ordinates  of  the  centre, 

J  ^  CE-2BD     ^  _  CB-'IAE 
^      4:AB-C'''  4:AB-C'' 

become  indeterminate  or  infinite,  according  as  the  numerators 
of  the  values  are,  or  are  not,  equal  to  zero...  In  both  cases  a 
change  of  origin  to  the  centre  (as  in  §  200)  is  impossible. 

In  the  case  where  h  and  h  are  indeterminate,  however,  the 
nature  of  the  locus  can  be  determined  without  any  transfor- 
mation of  co-ordinates  whatever. 
For  from  the  relation 

^AB-C'  =  0  (20) 

it  follows  that,  in  case  the  condition 

CE~2BB  =  0  (21) 

is  satisfied,  then  the  condition 

CD-2AE=0  (22) 

must  also  be  satisfied  {C  being  supposed  not  equal  to  zero). 
That  is,  the  numerators  of  h  and  h  must  vanish  together  ; 
whence  it  also  follows  from  equation  (3)  that  in  this  case  the 
condition  A  —  0 


212  ANALYTIC    GEOMETRY. 

is  satisfied.      Therefore    tlie    locus   must   consist   of  straight 
lines. 

Tlie  equations  of  these  lines  are  given  immediately  by  equa- 
tion (2),  which  may  now  be  written 

2%  +  Cx^E±,-jE''-^BF=  0.  (23) 

We  see  that  the  locus  consists  of  two  parallel  straight 
lines,  which  are  real,  imaginary,  or  coincident  according  as 
E'^  —  4  BF  is  positive,  negative,  or  zero. 

This  result  may  also  be  obtained  by  solving  the  general 
equation  with  respect  to  x,  and  then  introducing  the  condi- 
tions of  (20),  (21),  and  (22).     We  thereby  obtain  the  equation 

2Ax+Cij  +  D±:  ^~D'-^AF=  0.  (24) 

This  equation,  by  means  of  (20)-(22),  may  be  shown  to 
be  identical  with  (23). 

When,  in  addition  to  the  conditions  given  in  (20)-(22), 
E^  =  ^BF,  it  follows  that  i)'  =  4^i^  and  the  locus  is  the 
single  straight  line  represented  by  either  of  the  equations 

2Ax-\-Cij-\-D  =  0, 
2By+Cx+E  =  0. 

If  the  numerators  of  h  and  1c  are  not  zero,  the  centre  is  at 

infinity.     Although  we  cannot  now  transform  the  origin  to 

the  centre,  we  can  make  the  term  involving  xy  disappear  by 

proceeding  exactly  as  in  §  201  ;  that  is,  by  turning  the  axes 

through  an  angle  6,  the  value  of  which  is  determined  by  the 

equation  n 

^  tan  2  6)  =  —- (25) 

A-B  ^     ^ 

If  P,  Q,  U,  V  represent  the  new  coefficients  of  :r^  ?/',  x,  y, 
respectively,  P  and  Q  will  have  values  identical  with  those 
of  F  and  Q  given  in  §  201,  and 

U=     F  cose +  E sine,  (26) 

V=-FsmO  +  Ecose.  (27) 


LOCI    OF    THE    SECOND    ORDER.  213 

The  relations  of  T,  Q,  A,  B,  and  C\  found  in  §  201,  also 
hold  true,  namely, 

F+Q^A+B, 

P-B^±z  V(yl  -  By  +  C\ 

where  the  sign  before  the  radical  should  be  the  same  as  that 
of  C.     But  since  now  C^  =  ^AB,  the  value  oi  P—Q  become? 

P-Q  =  ±{A+B)- 

whence,  if  C  is  negative, 

P=0,     Q^A-\-B. 

But  if  C  is  positive, 

P  =  A+B,     Q  =  0. 

Suppose  that   C  is  negative.     Then  the  general  equation 

becomes  Qif +  Ux +  Vy -{-F=0.  (28) 

Divide  by  Q,  and  we  have 

•^      Q       Q'^     Q 


01"  y'  +  y.y  +  TTT^  +  ^P^+T^- 


or  [y  -\ = {  x-\ 

If  we  now  take  as  a  new  origin  the  point 
^QF-V V_ 

equation  (28)  becomes 

U 

which  represents  a  parabola  whose  axis  coincides  with  the 
axis  of  X,  and  is  situated  on  the  positive  or  the  negative  side  of 


214  ANALYTIC    GEOMETRY. 

the  new  origin,  according  as  U  and  Q  are  unlike  or  like  in 
sign  (§  103). 

The  vertex  of  the  parabola  is  the  new  origin,  and  the 
parameter  is  equal  to  the  coefficient  of  x  in  the  erpiation  of 
the  curve. 

Suppose   that    C  is    positive.     Then   the   general   equation 

^^^^^^^^  Px'  +  Ux  +  Yy  +  F=  0.  (29) 

And  this,  by  changing  the  origin  to  the  point 

4.PF-U'    _U_ 
4FP     '        2P 

becomes  x"-  =  — -y. 

This  represents  ^parabola  having  the  axis  of  y  for  its  axis, 
and  placed  on  the  positive  or  the  negative  side  of  the  new 
origin,  according  as  "p^and  P  are  unlike  or  like  in  sign. 

We  have  already  found  that  the  value  of  P  or  Q,  when  not 
zero,  is  ^  +  -^• 

We  may  obtain  the  values  of  U  and  V  in  terms  of  the 
original  coefficients,  as  follows  : 

From  (25)  we  find,  by  Trigonometry, 


Introducing  the  condition  ^AB  =  C^,  we  obtain 
tan  6  =  —  ^^'    if  C  is  negative  ; 

o  p 

=  ^^-^,        if  C  is  positive  ; 
G 

whence,  if  Cis  negative, 

sin^  = ^^  cos^=         ~^ 


V4^^+(7'^  V4^^  +  C 


'>2 
/ 


LOCI    OF    THE    SECOND    ORDER.  215 


And  if  C  is  positive, 


.    .  2B  .  C 

sin  6  — )     cos 


By  substitution  we  obtain  from  (26)  and  (27) 

^^2AE-CD^  (30) 

V=  ^^^^zM^,  (31) 

the  positive  sign  being  placed  before  both  radicals. 

204.  Special  Cases.  When  certain  of  the  coefhcients  in 
equation  (1)  are  equal  to  zero,  the  preceding  investigations 
may  require  some  modification  as  to  details,  not  as  to  results. 

Only  three  special  cases  recjuire  any  notice. 

1.  Suppose  that  B  =  0.  In  this  case  we  cannot  solve  the 
general  equation  with  respect  to  y,  and  hence  find  the  value 
of  A,  as  has  been  done  in  §  199.  If,  however,  we  solve  the 
equation  with  respect  to  x,  we  obtain 

^=-^^±^ViJ¥+W7+n;       (32) 

where 
X'  =  (7^-4ylA    J\r  =  2{CD-2AE),    N'=B'-4:AF] 

and  by  proceeding  as  in  §  199,  we  obtain  exactly  the  same 
value  of  A  as  before. 

In  this  case  2  cannot  be  zero ;  so  that  the  locus  always 
belongs  to  the  first  class  of  curves. 

2.  Suppose  that  A  =^  B  ^=^0.  The  general  equation  now 
becomes  Cxy  +  Dx  +  Ey  +  F=^0.  (33) 

In  this  case  the  value  of  A  cannot  be  found  directly  by  the 
method  of  §  199.     If,  however,  (33)  represents  straight  lines, 


216       •  ANALYTIC   GEOMETRY. 

the  equation  may  be  written  as  the  product  of  two  linear 
^'^^^^'^^^  {Px^3I){Qy-\-N)  =  0.  (34) 

Equating  coefficients  in  (33)  and  (34),  we  obtain 

PQ  =  C,   PN=D,    QM=E,   3IN=F, 
whence  CF=DE. 

When  this  condition  is  satisfied,  (33)  represents  two  straight 
lines,  and  their  equations  are 

Cx+E  =  0, 
Cy-\-D  =  0. 

The  two  lines  are  parallel  to  the  axes,  and  therefore  per- 
pendicular to  each  other. 

These  results  may  also  be  obtained  by  putting  A  =  0  and 
^  =  0  in  the  results  reached  by  the  general  investigation. 

In  general,  since  2  is  negative,  ecjuation  (33)  represents  an 
hyperbola.     By  changing  the  origin  to  the  centre,  the  equa- 
tion takes  the  form  ,      , 
ory  =  a  constant, 

which  we  know  (§  194)  represents  an  hyperbola,  referred  to 
its  asymptotes  as  axes.  Therefore,  in  general,  equation  (33) 
represents  an  equilateral  hyperbola. 

3.  Suppose  :S  =  0,  and  also  C=0.  Then  either  ^  or  ^ 
must  also  be  zero.  A  and  B  cannot  both  be  zero,  for  in  this 
case  equation  (1)  would  cease  to  be  an  equation  of  the  second 
degree. 

If  ^  =  0,  equation  (1)  becomes 

Bif  +  Dx-\-Ey-\-F=0, 

an  equation  of  the  same  form  as  (28).  Therefore  the  locus  is 
a  parabola. 

If  B=0,  equation  (1)  becomes 

Ax''-\-Dx-\-Ey-\-Fr^O, 

which  has  the  same  form  as  (29),  and  the  locus  is  a  parabola, 
with  its  axis  on  the  axis  of  y. 


LOCI    OF    THE    SECOND    ORDER. 


205.    The  main  results  of  the  investigation  are  given  in  the 
followincT  Table  : 


Loci  kepresexted  by  tue  General  Equation  of  the 

Second  Degree, 

.Ir-'  +  Bxf  +  Cxy  +  Dx  +Ey  +F=^0. 

CLASS.          1                 CONDITIONS. 

LOCUS. 

I. 

Loci 

having  a 

centre. 

2  positive,    A  not  zero. 
2  positive,    A  =  0. 
2  negative,  A  not  zero. 
2  negative,  A  =  0. 

Ellipse  (real  or  imaginary). 

Point. 

Hyperbola. 

Two  intersecting  straight  lines. 

II. 
Loci  not 
having  a 

centre. 

2  =  0,  A  not  zero. 
2  =  0,  A  =  0. 

Parabola, 

Two  parallel  straight  lines. 

Thus  it  appears  that  there  are  no  loci  of  the  second  order 
besides  those  whose  properties  have  been  studied  in  the  pre- 
ceding chapters. 

206.  Examples.  1.  Determine  the  nature  of  the  locus 
represented  by  the  equation 

Ix"  -llxy  +  6y'  +  23a:  -  2y  +  20  =  0. 

Here  we  have         S  =  — 121,  A  =  0. 

Therefore  the  equation  represents  two  intersecting  straight 
lines.  By  substitution  in  the  values  of  h  and  h  (§  200),  we 
find  that  the  lines  intersect  at  the  point  (2,  3). 

If  we  change  the  origin  to  the  centre,  the  equation  of  the 
lines  become 


218  ANALYTIC    GEOMETRY. 

2.  Determine   the  nature  of  the  locus  represented  by  the 
equation  ^x^  J^bf -\-2xy -12.x -I2y  =  0, 

and  reduce  the  equation  to  its  simplest  form. 

Here  2=96,     A  =  8G4,     A  =  1,     lc  =  l, 

^   E  =  12,     F=6,         Q  =  4. 

Therefore  the  locus  is  a  real  ellipse,  the  centre  is  the  point 
(1,  1),  and  the  equation,  in  its  simplest  form,  is 

The  value  of  0,  found  from  equation  (10),  is  45°. 
Therefore  the  equations  of  the  new  axes  of  x  and  y  referred 
to  the  original  axes  of  co-ordinates,  are  respectively 
x-y         =0, 

The  form  of  the  reduced  equation  shows  that    the    major 
axis  of  the  ellipse  is  situated  on  the  axis  of  y. 

3.  Determine  the  nature  of  the  locus  of  the  equation 

x-"  +  y'  -  5xy  +  Sx  -  20y  +  15  =  0, 
and  reduce  the  equation  to  its  simplest  form. 
In  this  case 

S=-21,     A  =-21,     h=-4,     k^O, 
i^=l,  P  =  -i       Q  =  h         ^  =  45°. 

Therefore  the  locus  is  an  hyperbola ;  and  since  H  agrees  in 
sign  with  Q,  the  transverse  axis  is  situated  on  the  new  axis 

of  y- 

The  equation  of  the  curve,  in  its  simplest  form,  is 
1y'-Sx'  =  2. 

And  the  equations  of  the  axes  of  the  curve  referred  to  the 
original  axes  of  co-ordinates  are 

x-y  +  4:  =  0, 
:r  -f  2/  +  -i  =  0. 


LOCI    OF    THE    SECOND    ORDER.  219 

4.  Determine  the  nature  of  the  locus  of  the  equation 

x"  ^  if  -  2x}j  +  2x  —  y  -l-=  0, 

and  reduce  the  equation  to  its  simplest  form. 

Here    :S  =  0,     A-1,     P=0,     Q  =  2.     ^  =  45°. 

Therefore  the  locus  is  a  parabola,  the  axis  of  which  coin- 
cides with  the  new  axis  of  x. 

V2 
From  equation  (30)  we  have  U—-—-' 

A 

Hence  the  equation  of  the  curve,  in  its  simplest  form,  is 

Since  C/^and  Q  agree  in  sign,  the  parabola  is  situated  on 
the  negative  side  of  the  new  origin. 

After  the  original  axes  of  co-ordinates  have  been  turned 
through  the  angle  -q  _  4^0 

the  vertex  of  the  parabola  is  the  point 

/  25       3V2\ 
V8V2'       8  -/ 
and  the  equation  of  its  axis  is 

3V2 

If  the  axes  of  co-ordinates  are  turned  through  the  anf^le 


to  their  original  position,  the  vertex  becomes  the  point 
/19     31\ 

and  the  equation  of  the  axis  becomes 

4:r-4y-f  3  =  0. 


220  ANALYTIC    GEOMETRY. 

207.  The  locus  of  a  point  which  so  movos  that  its  distance 
from  a  fixed  point  bears  a  constant  ratio  to  its  distance  from  a 
fixed  straight  line,  is  called  a  Conic. 

The  fixed  point  is  called  the  Pocus  ;  the  fixed  straight  line, 
the  Directrix  ;  the  constant  ratio,  the  Eccentricity. 

If  we  denote  the  distance  from  the  focus  to  the  directrix  by 
d,  the  eccentricity  by  e,  and  take  for  the  axis  of  x  the  perpen- 
dicular from  the  focus  to  the  directrix,  and  for  the  axis  of  y 
the  directiix,  the  equation  of  a  conic  is  easily  found  to  be 

f-\-{x-dy  =  e'x\ 
or  (1-  e>^  +  2/'  -  '2.xd-{-d'  =  0. 

This  is  an  equation  of  the  second  degree ;  hence  a  conic  is 
always  a  locus  of  the  second  order. 

If  e  —  0,  the  curve  is  a  circle  (§  70). 
If  c  <  1,  the  curve  is  an  ellipse  (§  162). 
If  e  =  1,  the  conic  is  a  parabola  (§  101). 
If  e  >  1,  the  conic  is  an  hyperbola  (§  195). 

In  many  treatises  the  properties  of  these  curves  are  deduced 
from  the  definition  above  given. 

208.  The  term  "  conic  "  is  an  abbreviated  form  of  "  conic 
section."  The  four  curves,  circle,  parabola,  ellipse,  and  hy- 
perbola, were  originally  called  conic  sections,  because  it  was 
discovered  that  they  could  all  be  obtained  by  making  a  plane 
cut  the  surface  of  a  cone  of  revolution  in  different  ways. 

If  the  plane  be  perpendicular  to  the  axis  of  the  cone,  the 
section  made  by  the  plane  will  be  a  circle. 

If  the  plane  be  parallel  to  an  element  of  the  surface,  the 
section  will  be  o,  parabola. 

If  the  plane  make  a  greater  angle  with  the  axis  than  the 
elements  make,  the  section  will  be  an  ellipse. 

If  the  plane  make  a  less  angle  with  the  axis  than  the  ele- 
ments make,  the  section  will  be  an  hyperbola. 


LOCI    OF    THE    SECOND    ORDER.  221 

Still  further  : 

If  the  plane  pass  through  the  axis,  the  section  will  consist 
of  two  intersecting  straight  lines. 

If  the  plane  pass  through  the  vertex  perpendicular  to  the 
axis,  the  section  will  be  s,  point. 

If  the  plane  be  parallel  to  an  element,  and  we  conceive  the 
vertex  removed  to  an  infinite  distance,  the  cone  will  become 
a  cylinder,  and  the  section  will  consist  of  two  parallel  lines. 

Hence  conic  sections,  in  the  most  general  sense  of  the  term, 
embrace  all  loci  of  the  second  order. 

Ex.  40. 

Determine  the  nature  of  the  following  loci,  and  reduce  each 
equation  to  its  simplest  form. 

1.  3:r'  +  2?/'-2:i:  +  y-l=0. 

2.  l+2:r  +  3?/^  =  0. 

3.  y"  -1xy-\-x^-%x^\<6  =  ^. 

4.  3.z-^  +  2r<;7/+3y^-16y  +  23  =  0. 

5.  a;^-10xy  +  3/'  +  5;  +  y  +  l  =  0. 

6.  rr^-2.ry  +  y'-6a;-63/  +  9  =  0. 

7.  a;2  +  .r?/+y'  +  ^'  +  y-5  =  0. 

8.  f-x'-y^^. 

9.  mx'  +  24.7:7/  +  29y^  -72:r  +  I263/  +  81=  0. 

10.  ?/2-2.r-8y  +  10-=0. 

11.  4a;'  +  9y^  +  8.r  +  36?/  +  4  =  0. 

12.  52;r'H-72:ry  +  73y'  =  0. 

13.  9y2-4:r'-8.r  +  18y  +  41  =  0. 

14.  y^  —  .ry  —  S.'T-f  5y  =  0. 

15.  16:^'  -  24.17/  +  9y'  -  Ihx  -  lOOy  =  0. 

16.  4.T^  +  4:ry  +  y'  +  8.T  +  4y-5--0. 


)  < 


ANSWERS. 


Ex.  3.     Page  7. 

1.  Let  Xj  =  —  2,  2/i  =  5,  x^  =  —  8,  2/2  =  —  3.  Substituting  in  [1],  we 
have 

d  =  V^-  ()/  +  (-  8)"^  =  vTuo  =  10. 

In  Fig.  3  the  points  Panel  Q  are  plotted  to  represent  this  case.  If 
we  choose  to  solve  the  question  without  the  aid  of  [1],  we  may  neglect 
algebraic  signs,  and  we  have 

QE  =  NO  -MO  =    S-   2=      6  ; 
FB  =  PM  +  i/i2  =    5  +    3  =      8  ; 
.-.  PQ2  =  QRi  +  PE^  =  3G  +  64  =  100,  and  PQ  =  10. 

2.  13.  8.   5,  5,  G. 

3. '5.  9.   a,  i,  Va^Ti^. 

4.  10.  10.    V29^,  2VI0,  4V5; 

5.  2\/a2TP.  ,2Vl0,  9^/2. 

6.  25,  29,  20V2.*^^^  11.   8  or  -16. 

7.  2\/r7,  5\/2,   VlOe.  12.   (x-  -  Tf  +  (y  -  2)^  =  121. 

13.    {x-%^  +{y-  3)2  =  {x-  4)2  +  (3/  -  5)2,  which  reduces  to  .r  4-  3/  =  7. 

Ex.  4.     PageN9. 

1.  (6,6).  3.   (2,-2).  ^  5.   (7,1). 

2.  (-1,  0).  4.  ^(3,  -1),  (*,"-V-),  (- J,  -f).       6.   (a,  -6). 
7.   Take  the  origin  of  co-ordinates  al  the  intersection  of  the  two  legs, 

and  the  axes  of  x  and  y  in  the  directions  of  the  legs.  Then,  if  a  and  b 
denote  the  lengths  of  the  legs,  the  co-ordinates  of  the  three  vertices  will 
be  (0,  0),  (a,  0),  and  (0,  h). 

10.  Observe  that  now  the  distances  RB  and  BQ  will  be  x  —  x^  and 
2^-2^^^         ^  12.   ill).  14.   (7i-31|). 

11.  A  2).  13.   (4,8).  15.  (13,-1). 


ANALYTIC    GEOMETIIY. 


Ex.   7. 

■  Page  23. 

1. 

12,  16. 

14. 

Locus  does  not  cut  the  axes. 

2. 

-10,6. 

15. 

(5,  7). 

3. 

±  4,  ±  4. 

16. 

(2,  1). 

4. 

±!.±2. 

17. 

(3, -4)  and  (-4,  3). 

5. 

±  I,  imaginary. 

18. 

(3,  4). 

6. 

±f,-4. 

19. 

(5,  3)  and  (3,  5). 

7. 

±  6,  ±  a. 

20. 

(0,  0)  and  (2,  4). 

8. 

3  on  OX. 

21. 

(5, -3),  (6,  4),  (-1,-4). 

9. 

+  3  on  OX 

22. 

\/6l,  V265,  VlO^L^.         ^ 

10. 

Locus  passes 

throu 

g^ 

origin. 

23. 

3^4;^^ 

11. 

Locus  passes 

throu 

gb 

origin. 

24. 

(a,  h){-a,  h),{-a,-h){a,- 

12. 

(  On  OX,  8, 
.On  OF,  2: 

and  - 

-4. 

25. 

No. 

4:\/48 

26. 

10. 

13. 

Locus  passes 

throu 

gl^ 

origin. 

h). 


Ex.  9,     Page  31. 

1.  Let  X  and  y  denote  the  variable  co-ordinates  of  the  moving  point. 
Then  it  is  evident  that  for  all  positions  of  the  point  x==Sy.  There- 
fore the  required  equation  is  x  =  Sy  or  x  —  3y  =  0.  Does  the  locus  of 
this  equation  pass  through  the  origin  ? 

2.  rr-6  =  0,  a: +  6  =  0,  x  =  0. 

3.  7/ -4  =  0,  y  +  l  =  0,  y  =  0. 

4.  The  line  x  =  3  is  the  line  AB  (Fig.  74);  how  is  this  line  drawn? 
The  locus  of  the  variable  point  consists  of  the  two  parallels  to  AB,  drawn 
at  the  distance  2  from  AB.  Let  CD,  EF,  be  these  parallels,  and  {x,  y) 
denote  in  general  the  variable  point,  then  for  all  points  in  CD  x  =  2> 
+  2  =  5,  and  for  all  points  in  EF  cc  =  3  —  2  =  1.  Therefore  the  equation 
of  the  line  CD  is  x -5  =  0,  and  that  of  the  line  ^i^  is  a;-l  =  0.  The 
product  of  these  two  equations  is  the  equation  (a;  — 5)  (a;  — 1)  =  0.  This 
equation  is  evidently  satisfied  by  every  point  in  each  of  the  lines  CD 
and  EF,  and  by  no  other  points.  Therefore  the  required  equation 
is  (x  — 5)  (a; -1)  =  0,  or  a;'^  — 6a;  +  5  =  0.  Verify  that  this  equation  is 
satisfied  by  points  taken  at  random  in  the  lines  CD  and  EF. 


ANSWERS. 


3 


5.  3/^  — 11  ?/ +  2i  =  0,  two  parallel  lines. 

6.  x^  +  8x  —  d  =  0,  two  parallel  lines.  7.   a: +  3=0,  y-2  =  0. 
8.   It  is  proved  in  elementary  geometry  that  all  points  equidistant 

from  two  given  points  lie  in  the  perpendicular  erected  at  the  middle 
point  of  the  line  joining  the  two  given  points.  This  perpendicular  is 
the  locus  required,  and  its  equation  evidently  is  x  =  3. 


Y 

F 

B 

D 

0 

E 

A 

X 

C 

Y 

P 

7)  A      X 


Fig.   74. 


Fig.   75. 


Let  us  now  solve  this  problem  by  the  analytic  method.  Let  0 
(Fig.  75)  be  the  origin,  A  the  point  (6,  0),  and  let  F  represent  any  posi- 
tion of  a  point  equidistant  from  0  and  A,  x  and  y  its  two  co-ordinates. 
Then  from  the  given  condition 

PO  =  PA, 

Therefore  x'^+y'^  =  {x-  Qf  +  {y-  Of, 

or  a;2  +  3/2  =  a;2-12a;-h36+y2. 

whence  rr  =  3, 

the  equation  of  the  locus  required. 
9.   a; -1=0.       10.    7/- 2-0. 


y- 


11.   x-y-l^O.       12.   x-y  =  0. 

13.  a;2  +  y2  ^  ]^00,  a  circle  with  the  origin  for  centre  and  10  for  radius. 

14.  Express  by  an  equation  the  fact  that  the  distance  from  the  point 
{x,  y)  to  the  point  (4,  —3)  is  equal  to  5.  The  equation  is  (.r  — 4)^ 
-f(2/  +  3)^  =  25. 

15.  (x  +  4)2  +  (7/ +  7)2  =  64.  16.   x2-i-y2  =  Sl. 

17.  Draw  ^0  ±  to  5C(Fig.  76).  Take  AO  for  the  axis  of  x,  and 
BC  for  the  axis  of  y ;  then  A  is  the  point  (3,  0). 

Let  P  represent  any  position  of  the  vessel,  x  and  y  its  co-ordinates 


4  ANALYTIC    GEOMETRY. 

OM  and  PM.     Join  PA,  and  draw  PQ  ±  EC,  and  meeting  it  in   Q. 
Then  from  the  given  condition 

PA  =PQ=^OM. 

Therefore  PA^  =  0M\ 

Now  PA^  =  AM''  +  PIP  =  (x  -  3)2  +  3/2,  and  OilP  =  a,.2.    Substituting, 
we  have 


{x-Zf-Vy-^ 


whence 


2/2  =  6a; -9. 

B 

P^ 

** 

(i 

0 

1       A       M 

X 

c 

"^ 

Fig.   76. 


The  locus  is  the  curve  called  the  parabola.     "We  leave  the  discussion 
of  the  equation  as  an  exercise  for  the  learner. 

18.  If  BCis  taken  for  the  axis  oft/,  and  the  perpendicular  from  A  to 
5Cas  the  axis  of  x,  the  required  equation  is  3/2  =  12cc  — 36. 

19.  x^  —  37/2  =  0,  two  straight  lines. 


20.  x2  +  1/2  =  JT^  —  a2,  a  circle. 

21.  4aa;  ±  ^2  ^  0^  two  straight  lines. 


Ex.  10, 


4.    cZ  =  \/] 


+  .71 


Page  33. 

6.  re  +  3/  =  7. 

7.  (¥.!);  fV2. 


5.   (x  -  4)2 -1- (3/ -  6)  =  G4. 

8.  Take  two  sides  of  the  rectangle  for  the  axes,  and  let  a  and  b  rep- 
resent their  lengths  ;  then  the  vertices  of  the  rectangle  will  be  the  points 
(0,  0),  (a,  0),  (a,  h),  (a,  b). 

9.  Take  one  vertex  as  the  origin,  and  one  side,  a,  as  the  axis  of  x; 
then  (0,  0)  and  (a,  0)  will  be  two  vertices.  Let  {b,  c)  be  a  third  vertex  ; 
then  (a  -i-  J,  c)  will  represent  the  fourth. 


ANSWERS.  5 

10.   (11,2).  11.   (5,-2),(^,-i).(|,-V)- 

12.   (1,  - 1).  13.    V^.         14.   (I,  I).  16.   (6,  23). 

1^-  1^       4       '  4      V    ^      2     '         2      J'    V      4       '  4       j' 

21.    3  or  ~  23.  23.    (4,  8)  and  (4,  -  8). 

1 3  and  2  on  OX.  24.    (2  a,  a)  and  (-  2  a,  a). 

16  and  1  on  or.  25.    (a,  0)  and  (-  a,  0). 

26.  10,  VIuI,  V52. 

27.  Taking  the  fixed  lines  for  axes,  the  equation  is  3/  =  6x  or  a  =  6?/. 

28.  Taking  A  for  origin,  and  AB  for  the  axis  of  x,  the  equation  is 
X  -  V3y  =  0. 

29.  Taking  the  fixed  line  and  the  perpendicular  to  it  from  the  fixed 
point  as  the  axes  of  x  and  y  respectively,  the  required  equation  is 
a?  —  ^y^  =  [y  —  af. 

Ex-  11.     Page  40. 

1.  x-y+l  =  0.  20.  2/ +  3  =  0. 

2.  2x-y-3  =  0.  21.  a;-2  =  0. 

3.  a;  +  y  - 1  =  0.  22.  a-  -  y  +  2  =  0. 

4.  a;  -  y  =  0.  23.  a:  -  y  +  5  =  0. 

5.  3a;  +  2y-12  =  0.  24.  a;-y-4  =  0. 

6.  2.r-3y  +  6  =  0.  25.  a;  -  VSy  -  4V3  =  0. 

7.  ^tc  +  y  -  7  =  0.  26.   y  +  4  =  0. 

8.  4a:-3y  =  0.  27.    V3a:-y-4  =  0. 

9.  y=0.  28.   a;  =  0. 

10.   y  =  4.  29.  \/3  a;  +  y  +  4  =  0. 

30.  a:  +  y  +  4  =  0. 

31.  a;  + V'3y +  4\/3  =  0. 
3  =  0.                                 32.  a:  -  y  -  4  =  0. 

14.  VSx-y +7-2\/3  =  0.  33.    3a;  +  4y-12  =  0. 

15.  a;  -  y  +  14  =  0.  34.   x-  -  3y  -  6  =  0. 

16.  VSx  +  Sy +12-13^3  =  0,  35.   aj  +  y +  3  =  0. 

17.  x-v/3y-^^0.  36.    3a; -5y -15  =  0. 

18.  a;  +  y-3=^.^  37.   a;-2y+10  =  0. 

19.  V3a;+y=0.  38.    a--y-l=0. 


6  ANALYTIC    GEOMETRY. 

39.  x-y-n  =  0.  52.  x  +  y -\- C^y/i 

40.  •l,T  +  3/-4n  =  0.  53.  (1,  7). 

41.  .r  +  y-5V2  =  0.  54.  (1,2). 

42.  X  -  y  V3  +  iO  =  0.  55.  (2,  1). 

43.  ,r+y\/3+ 10  =  0.  56.  (3,  2). 

44.  x-yy/2> -10  =  0.  57.  (-2,-6). 


45. 
46. 


a;+73/  +  ll  =  0,  a;-32/+l=0,  58.  (-4,6). 

3x  +  y-7  =  0.  59.  (5,  -  3),  (6,  4),  (- 4, -1). 

a; -7  3/ =  39,  9a;-5y  =  3,  60.  9a'  +  2?/  =  0,  \V^b. 

4x  +  2/  =  ll.  61.  y  —  X  =  yy^  —  x-^. 


(llx-'dy  =  25,  7a: +  9?/  r  (t?-c)a;-(6-a)3/ =  atZ~6c, 

1      =-17,  5.T-6y-21  =  0  *    \  {d-c)x  +  {h-a)y  =  hd-ac. 

■  5a;  - 3/  =  0,  5 a;  +  6?/ - 35  =  0,  f  23/2^7:  +  (a-^ -2x^)y  -  x^y^  =  0, 

48.  ■  3a;-2/  =  21,  9a;  +  4y  =  0,  64.    \  y^x  +  {2xi- x.;)y  -  Xiy2  =  0, 

^y  =  0,Ux  +  3y  =  2d.  [  y^x  -  {x^  +  x^)y=  0. 

49.  a;-y\/3-7*=-0.  65.   m  =  4. 

50.  y  =  a:  +  3.  66.   m  =  3. 

51.  a;  +  y-6\/2  =  0.  67.   6  =  -  9. 

6^-  !^  =  '?-^^^  ^^-^  ^i(y2-3/3)  +  ^2(y3-3/i)  +  ^3(yi-3/2)=o. 

3  1  2  1 

Ex.  12.  Page  45. 

1.  o  =  4,  6  =  7,  m  =  -  |.  12.  a  =  6  =  0,  m  =^  i 

2.  a  =  27,  6  =  -  9,  m  =  i  13.  a  =  &  =  0,  m  =  3. 

3.  a  =  -f,  6  =  1,  m  =  |.  14.    a  =  -  4,  6  =  5,  m=|. 

4.  a  =  J  =  0,  m  =  f.  15.    a  =  -2,  6  =  12,  7n  =  6. 

5.  a  =  6  =  0,  m  =  -|.  16.   a  =  6,  6  =  -  6,  m  =  1. 

6.  a  =  8,  6  =  -6,  m  =  f.  17.   a=-10,  6=-J/V3,  w=-^\/3. 

7.  a  =  6  =  3,  m  =  -l.  18.   a  =  10,  6  =  i^ V3,  m  =  i\/3. 

8.  a  =  -11,  6  =  --V,m  =  -i.  ^ 

20.   m  =  -^. 

9.  a  =  -3,  6  =  5,  m  =  -|.  a 

10.  a  =  2,  6  =  -3,  m  =  f.  21.   0 ;  -  8. 

11.  a=-2,  6  =  -3,  m  =  -|.  22.   ^  =  -  4,  ^  =  -  1. 


ANSWERS. 
24.    I  ^  ^  (3/2  -  yiX  ^  =  -  (-^2  -  .Ti),  25    ^  _  3/2 -.Vi    J  _  ^2^1  -  a^i3/2 

^  ^^  =  ^2yi  -  •1^2-  ■  •'•i-^'l'  ^-2-2^1 

27.    (a)  cos  a  =      j^,  sin  a  =      ^V  P  =  2. 

(6)  cos  a  =  —  jf ,  sin  a  =  —  /j,  p  =  2. 

(c)  cos  a  =      II,  sin  a  =  —  y\,  p  =  2. 

(c?)  cos  a  =  —  |-|,  sin  a  =      -j^,  p  =  2. 


Ex.  13.     Page  49. 

1.  3 a; -y- 16  =  0.  6.   .t  +  43/  +  49  =  0. 

2.  3a; -43/ -3=0.  7.    7.T-232/ +  193  =  0. 

3.  4a; -3/ =  0.  8.   y  =  2x.^ 

4.  3/- 8  =  0.  9.    7x  +  5y-ll  =  0. 

5.  a;- 5  =  0.  10.   a;  +  83/ -  5  =  0. 


n 


Ex.  14.      Page  51. 

2.   tan  d>  =  J.  3.   tan  (p  =  J^  •  4.    tan  A 

V      2  r      18  ^     w2  +  2 

5.   90°.  6.   45°.  7.  90°.  8.    0°.  9.   30°. 

11     (y  =  '^-'^-10,  j3     |y-3  =  m'(a;-2), 

■    la; +  53/ =  28.  '1      and  m' =  -  (5  ±  3\/3). 


12. 


3/  =  5a; +  11,  /  3/ —  3  =  m^(a;  —  1), 

l.  +  53/-3  =  0.  14.^      andm-^*^^- 


22.  2a;  +  3y-31  =  0.  '^  H 

23.  62a;  +  31 V -1115  =  0.  f    ^-3y+26  =  0, 

30.  <  5x  +  3y+    8  =  0, 


24.  v  =  6a; -27.  0,00      a' 
^                    ^  2x  +  3y—    9  =  0. 

25.  y  =  mx±d\/l  +  m\  ^^     a:_6  =  0 

26.  Bx  =  Aiy-b).  '        3.- '93/+    12  =  0. 

27.  ax-hy^f-h\  32.  J  10a;-   43/+    63  =  0. 


28.   (a±%-(aT6)(a;-a)  =  0.  U8a;-40y  +  111  =  0. 

{a;—    y—   6  =  0 -j  meeting  in  the  point 
2x~    y-   2  =  0  I  (-4,  -10). 

5a;_3y_10  =  oJ  Distance  =  V85. 

OK  -7l±5tand>,^      ^N 

35.    y  —  y,  = : (x  —  x,). 

-^      -^^        ^i^tand)    ^         '^ 


ANALYTIC    GEOMETRY. 

Ex.  15.     Page  57. 

|V5.  3.   4.  4.   yVVl3.  5.   0. 

The  learner  should  construct  the  given  line, 
and  observe  how  the  sign  of  the  required  dis- 
tance changes  when  we  cross  the  line. 

8.  6,  5,  4,  3,  2,  1,  0,  —1.  The  learner  should  construct  the  lines,  and 
observe  the  change  of  sign  of  the  distance,  as  in  No.  7.  A  study  of 
Nos.  7  and  8  will  make  it  evident  that,  in  equation  [12],  if  Ai\  +Bij^  +  C 
has  the  same  sign  as  C,  then  the  point  {x^,  y^)  lies  on  the  same  side  of  the 
line  as  the  orio-in,  and  vice  versa. 


1.  ^ViO. 

2. 

7.  ¥-,¥. 

¥> 

¥, 

1.  1. 

0, 

-I 

¥.¥. 

I 

f. 

1,   0, 

-I 

9. 

fViO. 

17. 

Va'  +  h\ 

22. 

4. 

10. 

IfVil. 

18. 

ah              3  ah 

23. 

,     C-C' 

11. 

2V2. 

y/d'  +  h'^     Va"^  +  6- 

y/A'  +  B' 

12. 

|VT8. 

-V- 

19. 

a2-62 

24. 
25. 

fVlO. 

13. 

Va^  +  62 

T3^^^- 

14. 

-AV13. 

-V-V2. 

4V2. 

20. 
21. 

Ah+Bk  +  C-D 

26. 

15. 

y/A'  +  ^2 
2. 

Zah 

16. 

2y/d'  -r  62 

Ex.  16.     Page  60. 


1.   IJ.                  4.   40. 

8.   35. 

11.   26. 

2.    12.                   5.   ah. 

9.    19.^. 

12.   96. 

3.   29.                   7.   26. 

10.  Kx^y2-x^/{). 

13.   61. 

14.  ^(a-c)(6-l). 

15.  l(a-h){2c-a-h). 

16.  Uo^-a^). 

21.  da'. 

22.  2^. 
21 

27.^. 
27n 

28.   },ah. 

17.  60°,  60°,  60°;  9  V3. 

18.  10. 

23.  24. 

24.  36. 

29.  ^^ 

2AB 

19.   }. 

25.    16. 

30.   56. 

20.   H.  26.    ^ah.  31.   10^. 


ANSWERS. 

Ex.  17.      Page  62. 
3.   2,  00,  90°,  2,  0°.  4.   0,  0,  45°,  0,  135°. 

2>/3-l 


5.   ^V3-2, 

2V3- 

-1,    60 

^   '^-^'    ^^^  • 

6.   I 

,    §V3,    150°,    1,    GO 

0_ 

26. 

4y  =  .r +  8. 

7.   1 

I,   -5v/3,    30°,    1,   - 

60°. 

27. 

4y  =  9a;-21. 

8.  I 
9. 

>/o,    -2,    G0°,    1,    - 
a;  +  23  7/-18  =  0, 
49x+    7y-82  =  0. 

30°. 

28. 
29. 

•9a;-203/  +  9G  =  0, 
.5x-    4y  +  32  =  0. 
88x-12l3/  +  371  =  0. 

10., 
11. 

-\>/82. 

•    3a;  +  43/-57  =  0, 

3a;  +  4y+    6  =  0, 

12^-52/ -39  =  0, 

30. 
31. 

5x-    y-10  =  0, 

.    x  + by -2^  =  0. 

2x  +    y-    9  =  0, 

,     a;-2y-17  =  0. 

12a; -5y  + 24  =  0. 

32. 

a;  ±  2/  -  5  =  0. 

area  =  63. 

33. 

2x  =  3/,   2y  =  r.. 

12.   4 

3. 

34. 

4a;  +  5y +  11  ±  3  vTl 

13.   a 

;  =  3. 

35. 

2/  =  (l±V2)(,r  +  2). 

14.    1 

x  -  ?/  + 1  =  0, 

36. 

13.y-12      23  +  5V29 

20. 


(x  +  y-l  =  0.  13a;-30  14 

15.  5x  +  6y-39  =  0.  37     f  7a;~32/  + 15  =  0, 

16.  14a;-3y-30  =  0.  *   I  3a;  + 72/- 19  =  0. 

17    4a:-5v  +  8  =  0  38.    |    8.' +  72/-19  =  0, 
Lt.   ^x     02/  +  5     u.  Il6x'  +  3y  +  17  =  0. 

18.   a: +  2/ -7-0.  39^   450^ 

19.  y-y^-v^-yx.  4o.  90°. 


41. 


31n/26 


7/  =  3,    13.y  =  5x-l,  ^-      143 


"51.   92a' +  692/ +  102  =  0.  '   \/d' 

22.   a; +  4?/ =  34. 


43. 


23.   3a;  +  42/- 5a  =  0.  Vli^  +  k^ 

24.-  3a; +  42/ =  24. 


44.  -Vl  +  m2. 


25.  3/-2/i  =  --|;(^-^'^i).  45, 


m 


/t^ 

54. 

b 

2a2  +  5a6  +  262 

57. 

6 

58. 

172^ 

59. 

59. 
60. 

no.  5R 

61. 

10  ANALYTIC    GEOMETRY. 

46.   _.  54.    xy  represents  the  two  axes. 

a  =  5. 

X  +  a  =  0,   X  —  h  =  0. 

X  +  a=-0,   y  +  b  =  0. 
KQ    rg  o^'   The  axes  and  x  =  y. 

51.   (10,  5|).  61.    2a; -2/ =  0,    7a;+2/  =  0. 

62.  If  h  denotes  the  altitude  of  the  triangle,  and  the  base  is  taken  as 
the  axis  of  a;,  the  locus  is  the  straight  line  y  =  h. 

63.  The  equation  of  the  locus  is 

{x  -  x,f  +  {y  -  y,f  =  {x-  x,f  +  (y  -  y,)^ 
This  is  the  equation  of  the  straight  line  bisecting  the  line  joining  (x^,  3/^) 
and  {x.^,  7/2),  and  ±  to  it. 

64.  The  two  parallel  lines  represented  by 

Ax-\-By  +  C±  c^VZM-"^  =  0. 

65.  x^y  =  h  66.   A^±By±C  _^A^x  +  B^y  +  C^  ^j^^ 

y/A^-\-B'  y/A'HB'-' 

67.  Let  h  denote  the  base,  k"^  the  constant  difference  of  the  squares  of 
the  other  two  sides.  Taking  the  base  as  axis  of  x,  and  middle  point 
of  the  base  as  origin,  the  equation  of  the  locus  is  2bx=  ¥■. 

Ex.  18.     Page  70. 

1.  7a?  +  7/  =  0. 

2.  a;  +  2.y  +  13  =  0. 

3.  5a; +  6?/ -37  =  0, 

9.  {AC'-A'C)x^i^BC'-B'C)y  =  ^. 

10  {BA' -AB')y  ^CA'~AC^  =  ^. 

11  ^.^  +  By  +  C  ____  A'x  +  B'y  +  C' 
^,Ti  +  %i  +  (7     A'x^^B'y^^& 

12.   472a; -291/ +174  =  0.  13.   3/ =xV3  +  3  -  VS. 

j4    I  4a; +  32/ -25  =  0,  y     x  _m'b-a 

'    1  3a;-4y+ 25  =  0.  "a      h      ma  +  b 

16-18.  Generally  the  easiest  way  to  solve  such  exercises  as  these  is 
to  find  the  intersection  of  two  of  the  lines,  and  then  substitute  its  co- 
ordinates in  the  equation  of  the  third  line. 


4     (a;-y  +  8=0, 
\  x  +  y-6  =  0. 

6. 

64  a; -233/ =  59. 

7. 

Ux+y  =  0. 

5.     y  =  x  +  3. 

8. 

5a; +2/ -16  =  0. 

ANSWERS.  11 

to             1                                              OA     -JIM         vi^^—m        1/^—h 
19.   m  =  l.  20.    uheii — -  = 

21.  If  we  choose  as  axes  one  side  of  the  triangle  and  the  correspond- 
ing altitude,  we  may  represent  the  three  vertices  by  (a,  0),  (— c,  0),  (0,  /;). 

22.  Choosing  as  axes  one  side  and  the  perpendicular  erected  at  its 
middle  point,  the  vertices  may  be  represented  by  (a,  0),  (—a,  0),  (&,  c). 

23.  It  is  well  here  to  choose  the  same  axes  as  in  No.  21. 

24.  Choosing  the  origin  anywhere  within  the  triangle,  it  is  evident 
that  the  equations  of  the  bisectors  in  the  normal  form  may  be  written  as 
follows : 

{x  cos  a-\-  y  sin  a—p)  —  {x  cos  a^  +  y  sin  a^  —  p^)  =  0, 

(x  cos  a^  +  y  sin  a^  —p^)  —  (x  cos  a^^  +  y  sin  a'^  — i>^0  =  0. 
(x  cos  a^^  +  y  sin  a^^—p^^)  —  (x  cos  a  -f  y  sin  o  —p)      =  0. 
Now,  by  adding  any  two  of  these  equations,  we  obtain  the  third ; 
therefore  the  three  bisectors  must  pass  through  one  point. 


25. 

2V2,    VIU,   2VW. 
.  Origin  within  the  A. 

29. 

J  a:  +  y  -  2  =  0, 

26. 
27. 

.uvio,  ifVsi,  i|Vl3. 

r  rr  +  3/  +  79  =  0, 
I  7x-1y  +  23  =  0. 

30. 

.r  - 1  =  0, 
ly-l  =  0. 

^8 

7x-9y  +  S4:  =  0, 
Ox +  72/ -12  =  0. 

31. 

2/  —  mx  —  b         y  —  m^x  —  b^ 

Vl  +  w^              Vl  +  m'2 

Ex.  19.     Page  74. 

1.  (i.)  Parallel  to  the  axis  of  x,  (ii.)  parallel  to  the  axis  of  y. 

2.  When  ad  =  be. 

3.  The  two  lines  are  real,  imaginary,  or  coincident,  according  as 
H'^  —  4AB  is  j^ositive,  negative,  or  zero.  The  two  lines  are  ±  to  each 
other  when  A  +  B  =  0. 

5.  a;  +  y  +  1  =  0,  and  x  —  3  y  +  1  =  0. 

6.  .T-2y  ±(y-3)  V^1  =  0. 

7.  .r-y-3  =  0,  and  x-Sy +  3  =  0.  8.45°.  9.    ir=  2. 
10.  Jr=-10  or  --V-.                   11.    ^=28.                   12.    /r=  V- 


12  ANALYTIC    GEOMETRY. 

Ex.  20.     Page  76. 

1.  Take  the  point  0  as  origin,  and  the  axis  of  3/  parallel  to  the  given 
lines.  If  the  equations  of  the  given  lines  are  x  =  a,  x  =  b,  and  if  the 
slopes  of  the  lines  drawn  in  the  two  fixed  directions  are  denoted  by 
m^,  m^^,  the  equation  of  the  locus  is 

(b  —  a)y  =  m^b{x  —  a)  —  m^^a{x  —  b). 

2.  If  a  and  b  are  the  segments  of  the  hypoten,use  made  by  a  perpen- 
dicular dropped  from  the  vertex  of  the  right  angle,  the  equation  of  the 
locus  is  y^xJi 

3.  Let  OA  =  a,  OB-=h.    Then  the  equation  of  the  locus  is  a- +  y  =  a  + 5. 

4.  Take  as  axes  the  base  and  the  altitude  of  the  triangle.  Let  b 
denote  the  base,  a  one  segment  of  the  base,  h  the  altitude.  *Then  the 
equation  of  the  locus  is  Ox        2?/ 

b  —  a       h 
This  is  a  straight  line  joining  the  middle  points  of  the  base  and  the 
altitude. 

5.  Take  as  axes  the  sides  of  the  rectangle,  and  let  a,  b  denote  their 
lengths.     The  equation  of  the  locus  is 

bx  —  ay  =  0. 
Hence  the  locus  is  a  diagonal  of  the  rectangle. 


Ex.  21. 

Page  79. 

1. 

x''  +  y^  =  -2rx. 

13. 

(4.  0),  4. 

2 

xUy^=2ry. 

14. 

(-4,0),  4. 

3. 

x^  +y^=-2ry. 

15. 

(0,  4),  4. 

4. 

(a;  _  5)2  +  {y  +  3)2  =  100. 

16. 

(0,-4),  4. 

5. 

x'  +  (y  +  2y^l2h^ 

17. 

(0,  1),  I 

6. 

^x  -  5)2  +y^  =  25. 

18. 

(0,0),  Sk. 

7. 

{x  +  5)2  +  2/2  =  25. 

{x  -  2)2  +  {y-  3)2  =.  25. 

19. 
20. 

(0,  0),  2  k. 

8. 

(0,  0),  Vd'  +  62. 

9. 

x^+y^-2hx-2Jcy  =  0. 

21. 

(|0)JV5. 

1 1 

(1,  2),  Vs. 

(i  1).  w^. 

22. 

Ll. 

L2. 

fh  k\   Vh^  +  k'' 

ANSWERS.  13 

23.  When  D^T)'  and  E=E^  \  in  other  words,  wlion  the  two  equa- 
tions differ  only  in  their  constant  terms. 

24.  In  this  case  r  =  0.  Hence  the  equation  represents  simply  the 
point  (o,  h).  We  may  also  say  that  it  is  the  equation  of  an  infinitely 
small  circle,  having  this  point  for  centre. 


r{i.)    C+and=i)2^ 

31. 

\  (ii.)  C+and  =E\ 

I  (ill.)  C  +  and  >  D^  and  >  E'^. 

32. 

a;^  +  y2_i0a; -102/ +  25  =  0. 

33. 

(7,  4)  and  (8,  1). 

34. 

(2.  0)  and  (f.  -f). 

35. 

|V5. 

36. 

V     d'  +  b-'l 

37. 

2a; -2/ -2  =  0. 

38. 

4:x-5y  +71  =  0. 

39. 

3x-5y-34:  =  0. 

26.  ^  On  OX,  3  and  2 ; 
I  On  OY,  6  and  1. 

r(0,  2),5; 

27.  I  On  OX,  fi  ±  V2I ; 
I  On  0  Y  imaginary  points. 
.(2,  4),<;/5; 

28.  ]  On  OX,  0  and  4  ; 
(  On  0  Y,  0  and  8. 

(-(3,-2).  3; 

29.  ^  On  OX,  7and-l; 
i  On  0  r,  -  2  ±  VT3. 
/-(-ll,  0),  VRT); 

30.  ]  On  OX,  -  3  and  -  19  ; 
iOn  or,  9±2v/(i. 

40.  Let  (.r,  y)  be  any  point  in  the  required  locus  ;  then  the  distance 
of  (x,  y)  from  {xi,  y^  must  always  be  equal  to  its  distance  from  (0^2,  y^) ; 
therefore  {x  -  x^f  +  (3/  -  y^f  =  {x  -  x^f  +  (y  -  3/2)2 . 
whence             2x{x^-  'x.,)  +  2y(y^-  y.^)  =  {x^^-  +  yi^  -  x,;'  -  y.^^). 

Show  that  this  represents  a  straight  line  ±  to  the  line  joining  [x^,  yj 
and  {x^,  y^). 

41.  8a: +  6y +17=0. 

42.  First  Method.  Substitute  successively  the  co-ordinates  of  the 
given  points  in  the  general  equation  of  the  circle ;  this  gives  three  equa- 
tions of  condition,  and  by  solving  them  we  find  the  values  of  a,  6,  r. 

Second  Method.     Join  (4,  0)  to  (0,  4)  and  also  to  (6,  4)  by  straight 
lines,  then  erect  perpendiculars  at  the  middle  points  of  these  two  lines ; 
their  intersection  will  be  the  centre  of  the  circle,  and  the  distance  frojtn  ^u 
the  centre  to  either  one  of  the  given  points  will  be  the  radius.  /J^^;^lt»t*  r     -^ 

43.  a;2  +  y2  -  8  ;c 

44.  x^  +  y2  +  G  X 


Ans.  a;2  +  y2  -  6 a;  -  By  +  8  =  0. 

6y  =  0. 

45.   a;2  +  y^  +  8  a.-i;  -  6  ay  =  0. 

y-0. 

46.    .T2+y2  +  8a-  +  20y +  31  =  0. 

14  ANALYTIC    GEOMETRY. 

47.    x-2-i-3/2_9a'-53/  +  14  =  0.  51.    x'-vy'-2ax 

48     10^-    5)2  +  (y  +  8)2  =  169,  52.   x'' +y^  =.ax +  hy. 

■    I  (.T  -  22)2  ^ty_  9)2  =  169. 

^  ^       vy        ;  g3    (a,_i)2^/       4^2_20. 

•  la;2+2/2-    6(a;  +  2/)+      9  =  0.       54.    rt^ +3/2_14a;-4y-5  =  0. 

50.  a'2  +  2/2-8a;-8y  +  16  =  0.  55.   x-^ +  y2  +  20?/ =  0. 

56.  ???.  (x^  +  ?/2)  4-  aZ>  =  (??ia  —  h)x  +  (mi  —  a)y.     bl .    x^  +  w^  =  x^.^  +  y^y. 

58.  (.r  -  .r,)  {x  -  x^)  +  (y  -  y,)  {y  -  y,)  =  0.        g^    ^,,  _  ^^,  ^  ^.,  ^  ^,,  _  a^ 

59.  (1  H-  ?«-)  (x2  +  2/2)  -  2  r  (x  +  my)  =  0.  ^ 

Ex.  22.      Page  87. 

1.    The  double  sign  corresponds  to  the  geometric  fact  that  two  tangents 
having  the  same  direction  may  always  be  drawn  to  a  given  circle. 
3.    2a;  +  3y  =  26,  Sx-2y=^0,   VlTY,    V;12,  9,  4,  -^S-VlS. 

4    L,    J. 

5.  9x-13?/  =  250. 

6.  x±?jy^lO.  \/ 

7.  104i 

8.  ic2  +  ^2_25. 

9.  14x±62/  =  232. 

10.  3x  +  ?/  =  19. 

11.  3x-42/  =  0. 
12     f3.T  +  72/  =  93, 

l3;y_7y  =  65. 

13.  x  =  r. 

14.  ^x  +  %  ±  r\/^2+^2  _  0.  31.    (x-  _  5)2  +  (y  -  3)2  =  -L2JL. 

15.  i?.K-^2/±rVJ.2+^2_o.  gg     |(a^-    2)2  +  (3/ -    4)2  =  100, 


22. 

1  (p  COS  a,  p 

2 

sin  0). 

23. 

■  When  (72  = 
When  r     = 

^.r  +  By  +  C 
V.-i2  +  ^2 

24. 

ax  +  hy  =  0. 

25. 

(-  a,  -  h). 

26. 

(2  a,  6). 

27. 

(0,  h). 

28. 

.t2  +  2/2  =  |. 

29. 

m  =  ±f. 

30. 

c  =  28  or  - 

52. 

16.  x-3/±rV2  =  0.  I  (x  -  18)2  +  (7/ -  16)2  =  100. 

17.  The  equation  of  the  two  tan-  33.   (,r  - 1)2  +  (y  _  0)2  =  25. 

gents  is  {h^—r^)y'^=r^{x  —  Tif.  Ill 

18.  x  +  y  =  r  V2.  '  r2  ~  a2  ^  62' 


jg     {x  =  10,  35.  (a2+62)(x2  +  2/2)-2a6Va2-i-62 
1  3 a-  +  43/  =  oO.  X  {x  +  2/)  +  a^6^  =  0. 

20,  y  =  2 X  +  13  ±  GV5.  36.  x  =  a  +  r. 

21.  21,  3«-.  37.  [4r2-2(a-6)2]5. 


ANSWERS.  15 

X.  Ex.  23.     Page  90. 

1-   (1,  -^).  2^29-  5-   •'■-'  +/  =  81. 

2.  K2  ±  V7),  i  Vn.  6.   (.r  -  7)2  +3/2  =  9. 

3.  (i,  I),  V-^.  7.   {X  +  2)2  +  (y  -  5)2  =  100. 


b  ah 


8.    ,r2+y2_2a(3a;  +  4y)  =  0. 


VVl  +  a-     Vl  +  a2 
9.   a;2  +  y2  +  262  +  c2  =  2[(&  +  ^^^^  +  (j  _  c),/]. 

10.  3  ah{x^  +  2/^)  +  2  a6(a2  +  h"")  =  (5  o2  +  o  ^2)60;  +  (5  62  -^  2  a2)ay. 

11.  .T2^2/2-5x-12y  =  0.  13.   x2  +  2/2  +  14a; +U2/  + 49  =  0. 

12.  a;2  +  3/2  -  14a;  -  43/  -  5  =  0.  14.    a-  +  7/2  _  2ra;  -  2 ?'?/  +  r2  =  0. 

15.  a;2  +  3/2  _  2a.r  -2ay  +  3a2--  =  0. 

16.  a.'2  +  3/2  =  |.  17 .    5(a'2  -\- 1/)  - 1 0  .t  +  30  3/  +  49  =  0. 

[  (a;  _  -y-)2  +  (y  _  Y)2  =  ^.  20.    a;2  +  y2  +  50  a;  +  88  y  +  230  =  0. 

21     f       ^•'+       y2_3(3^^_    46y  + 324  =  0, 
(  25  a;2  +  25  y2  -  80  a-  -  494  y  +    64  =  0. 

22.  (6,2),  5.  25.   iV234. 

23.  15.  28.    VIO. 

24.  10.  27.    x^x  +  y^y  =  x^^  +  yi2. 

28.  (i.)Z)2=4^(7^  (ii.)^2_4^C'_  (m.)Z)2=^2_4^(7_ 

29.  r2  =  2rmc  +  c2.  35.  6.6- -  8y  -  25  =  0. 

30.  ;>;  =  40.  36.  2(37  ±  3  V41)x- +  25y  =  0. 

31.  a2+y2  =  ay\/2.  37.  a; +  V3y±  20  =  0. 

32.  a;2  +  y2  ±  2a{x  ±  y)  =  0.  38.  a;  +  y  -10  =  0. 

33.  2(a;2  -  ax  +  y2  -  ?-2)  +  a2  =  0.  40.  .](35  +  24  V30). 

34.  x-y  =  0.  43.  135°. 

|-  ix  4-  4)2  +  (y  +  10)2  _  85.  44.    (7,  -  5)  and  (-  63^  92-6). 

45.  \  I        oUy  ,  /      ,  670\2      85 


U'^      169;  ^(,^"^169  j      1692' 

46.  The  circle  {x  —  x\f  +  (y  —  yi)2  =  r^. 

47.  The  circle  {x  -  a)2  +  (y  -  6)2  =  (r  +  r'f. 

48.  The  circle  (a;  -  aj-  +  (y  ~  hf  =  r^  +  t^. 


16  ANALYTIC    GEOMETRY. 

49.  Take  A  as  origin,  and  let  the  radius  of  the  circle  =  r ;  then  the 
locus  is  tlie  circle  x^  +  y"^  =  rx. 

50.  Take  A  as  origin,  and  let  the  radius  of  the  circle  =  r ;  then  the 

2,mrx 
locus  is  the  circle  sf  -\-  y^  =  — ; — 

51.  Take  A  as  origin,  AB  as  axis  of  x,  and  let  AB  =  a  ■  then  the 
locus  is  the  circle  (m^  +  7^"')  (x^  +  y"^)  -2am?x  +  aHi^  =  0. 

52.  Take  AB  as  the  axis  of  x,  the  middle  point  of  AB  as  origin,  and 
let  AB  =  2a  ;  then  the  locus  is  the  circle  2{f  +  y"^)  =  Ic^  -2a}. 

53.  Using  the  same  notation  as  in  No.  52,  the  locus  is  the  straight  line 
4  ax  =  P. 

54.  Taking  the  fixed  lines  as  axes,  the  locus  is  the  circle  "^{x"^  ^- y"^)  =  d^ . 

55.  Take  the  base  as  axes  of  x,  its  middle  point  as  origin,  and  let  the 
length  of  the  base  =  2a,  and  the  constant  angle  at  the  vertex  =  d.  Then 
the  locus  is  the  circle  x^  +  ?/2  —  2a  cot  ^ 3/  =  a-. 

56.  Take  A  as  origin,   AB  as  axis  of  x,  and  let  AB  =  a,  AC=h. 

52 
Then  the  locus  is  the  circle  (x  -  a)^  +  2/^  =  —  • 

57.  The  circle  x^  +  v^  =  _ll where  I  is  the  length  of  the  chord. 

4  ^2  _  ^2 

58.  The  locus  consists  of  the  two  circles  x^  +  y^  ±  rx  =  0. 


Ex.  24.     Page  103. 

1.    7a;_63/  =  0.  2.   2x-2y  +  9  =  0. 

4.  x+2/  =  r,  2x  +  oy  =  r,  {a  +  h)x  +  {a  -  b)y  =  r-^. 

5.  13x  +  22/  =  49.  6.    The  tangent  at  (A,  ^). 

7.  (i.)  2x  +  3  3/  =  4,  (ii.)  3x-3/  =  4,  (iii.)  x  -  3/ =  4. 

8.  (i.)  (20,  30),  (ii.)  (21,  -14),  (iii.)  (35  a,  356). 

9.  (6,  8).  18.   12x  +  17y-51  =  0. 

10.  (-'^''l    -^\  19-    x  +  y-2  =  0. 

^  , , 20.  (a2-a6)x-(a6-62)y  +  ac  =  0. 

11.  (4,  ±V-12),  4x4:V-122/  =  4.  ^  '         ______ 

16    A2  +  F-r2.  2^-  a:-2/  =  0,  Vata+6/^-4c. 

17.    3.   '  *  22.  (-2,-1). 


ANSWERS.  17 

Ex.  25.     Page  115. 

1.  Writing  a;  +  i  for  x,  und  y  —  2  for  y,  ami  reducing,  we  have  y"^  =  4a;. 

2.  x'  +  2/'  =  r\  6.    x'  +  y-'  =  r\ 

3.  a;*  +  .V^  =  2  rx.  6.    2  xy  =  a?. 

4.  a;2+y2__2ry.  7.   a;2_3/2  =  2. 

8.  (i.)  p  =  a,  (ii.)/)2cos20-a2. 

9.  (i.)  p  =  4acot^  COS0,  (ii.) /!)(1  — cos^)  =  2a. 

10.  (i.)  x^  +  2/^  =  a^,  (ii-)  a;"^  +  y^  =  «a;.  (iii-)  -^'^  —  y"^  =  o?- 

11.  re  +  y  =  0.  15.   a;^  —  G  ary  +  3/^  =  0. 

12.  2a;-5y  =  0.  16.   a-?/ =  3. 

13.  12a;2  +  16  a-y +  43/2  =  1.  17.    y^  =  2a(a;V2  -  a). 

14.  a;2+y2  =  25.  18.   4ary  =  25. 


Ex, 

.  26. 

Page  117. 

1. 

6V3. 

12. 

dx^  +  2dy^  =  225. 

2. 

4  sin  J  ft). 

13. 

p  =  Sacosd. 

3. 

\/l3-12cosft,. 

14 
15. 

p  =  ia. 

4. 

Va"^  +  b'^-2abcoii{0- 

7). 

p  =  f  csc^^e. 

5. 

2  a  sin  6. 

16. 

p  =  49  sec  2^. 

6. 

2 a  cose. 

17. 
18. 

p  =  k^  cos  26. 

7. 

aVo  -  2 V3. 

xy  =  a^. 

9. 

2a;2  +  2a;y  +y^  =  l. 

19. 

{x^+y^)^  =  2kxy. 

10. 

2a;2  +  y2  =  6. 

20. 

ar'  +  y3  -  5  ^a;y  =  0. 

11 

y  =  0. 

21. 

tan-^ }. 

22.   (i.)tan-i('- 

-i\ 

(ii.)tan-^4 

Ex.  28.     Page  124. 

2.   y2  =  ipx  -  4p2.  3.   y2  =  ipx  +  \f. 

4.  (i.)y2  =  10a;,  (ii.)y2  =  10a;  +  25,  (iii.)  y^  =  lOx  -  25. 

5.  (i.)y2=16a;,  (ii.)  y^  =  16a;  +  64,  (iii.)  y^  =  16  a:  -  64. 


18  ANALYTIC    GEOMETRY. 

6.  (2,  ±G).  8.   (4,6)  and  (25,15). 

7.  6,  15,  ^.  9.    (12,6). 

0 

10.  The  line  x  =  9  meets  the  parabola  in  (9,  6)  and  (9,  -  6).  The 
line  x  =  0  passes  through  the  vertex.  The  line  x  =  —  2  meets  the  para- 
bola in  the  imaginary  points  (—  2,  ±  V—  8). 

11.  The  line  3/  =  6  meets  the  parabola  in  (9,  6).  The  line  y  =  —  8 
meets  the  parabola  in  (16,  —  8). 

12.  p  =  4.  13.    The  point  (2,  8). 

14.  (i.)3/  =  0,  (ii.)a;  =  -2,  (iii.)  a;  -  2,  (iv.)  x±y- 2  =  0,  {v.)y  =  2x. 

15.  (i.)  4a;  -  57/  +  24  =  0,  (ii.)  a:^  +  y^  =  20.r.      16.    3;^.      17.    S^/3p. 

26.  The  common  latus  rectum  =  4j3.  The  common  vertex  is  at  the 
origin.  The  axis  of  :r  is  the  axis  of  (i.)  and  (ii.) ;  that  of  y  is  the  axis  of 
(iii.)  and  (iv.).  Parabola  (i.)  lies  wholly  to  the  right  of  the  origin, 
(ii.)  wholly  to  the  left,  (iii.)  wholly  above,  (iv.)  wholly  below.  AVe  may 
name  them  as  follows  :  — 

(i.)    is  a  right-handed  X-parabola. 
(ii.)   is  a  left-handed  X-parabola. 
(iii.)  is  an  upward  I^'-parabola. 
(iv.)  is  a  downward  F-parabola. 

27.  (i.)  If  A  =  0,  the  equation  becomes  y^  +  By  +  0=  0,  a  quadratic 
in  y,  and  representing  the  two  straight  lines,  parallel  to  the  axis  of  x, 

T) 

whose  equations  are  y±~±  ^VB^  —  4  AC=-  0.    (ii.)  If  5  =  9,  the  origin 

A 
is  in  the  axis  of  the  curve,     (iii.)  If  C=  0,  the  origin  is  in  the  curve. 
(iv.)  If  J.  =  .S  =  0,  the  locus  consists  of  the  two  parallels  y  =  ±  V— C. 
(v.)  \i  A=  (7=0,  the  locus  consists  of  the  straight  lines  y  =  0,  y  +  B  =  0. 
(vi.)  If  B=  C=  0,  the  vertex  is  at  the  origin. 

28.  Latus  rectum  =  —  B. 

A     A^-^C 


The  vertex  is  the  point 

The  focus  is  the  point  ( 

The  axis  is  the  line  x  =  - 

2 

The  directrix  is  the  line  y  = 


2  45 

A  A^_-±C_B 

2'  45          4 
A 

A^  +  B'--iC 

4:B 

If  B  is  negative,  the  curve  is  above  the  axis  of  x. 
If  B  is  positive,  the  curve  is  below  the  axis  of  x. 


ANSWERS. 


19 


Latus 
Rectum. 

Vertex. 

Focus. 

Axis. 

Directrix. 

29. 

12 

(7,0) 

(10,  0) 

2/  =  0 

x  =  i 

30. 

12 

(-  7,  0) 

(-4,0) 

y  =  o 

a;  =  -10 

31. 

-12 

(-  7,  0) 

(-  10,  0) 

y  =  o 

.T  ■-=  -  4 

32. 

-12 

(7,0) 

(4,0) 

y  =  o 

a- =  10 

33. 

12 

(0,7) 

(0,  10) 

a;  =  0 

y  =  4 

34. 

8 

(6,4) 

(8,4) 

.V  =  4 

.r  =  4 

35. 

-f 

(-^0) 

(-iO) 

2/  =  0 

x  =  -i 

36. 

1 

(^.-f) 

(^.  -  2) 

x  =  h 

2/  =  -f 

37. 

4 

(-2,-3) 

(-1.-3) 

y  =  -3 

a;  =  -3 

Ex.  29.     Page  129. 

2.  On  the  axis  of  x  lay  off  from  the  vertex  to  the  left  a  length  equal 
to  the  abscissa  of  the  point  of  contact ;  this  determines  a  second  point  in 
the  tangent.  Lay  off  from  the  foot  of  the  ordinate  of  the  given  point, 
towards  the  right,  a  length  equal  to  2p  ;  this  determines  a  second  point 
in  the  normal. 

6.  x-4y +  20  =  0,  4a;+y-90  =  0. 

7.  Tangents  I  ^-3/ +  3  =  0,      normals  | '-^  +  ^  "  ^  ==  ^' 

^  [a;  +  .y  +  3  =  0;  \x-y-d  =  0. 

These  lines  enclose  a  square  whose  area  =  72. 

8.  Tangent  =  V266,  normal  =  V95,  subtangent  =  14,  subnormal  =  5. 
P 


9.   (5,  10). 


13. 


14. 


x,x. 


r'2. 


mv  1  +  m^ 
X  —  y  +  p  =  0,  point  of  contact  {p,  2p),  intercept  =  p. 


2  (yi  +  2/2)]- 


15. 

16.  Equation  of  the  tangents  yVS  =  ±x±3p,  required  point  (—  3p,  0). 

17.  For  the  two  points  whose  co-ordinates  are 


=  ^(1 
8^ 


Vl7), 


y 


^p\ 


I  +  VfT 


18.  For  the  points  (0,  0)  and  {3p,  ±  2pV3). 

19.  9.r-6y  +  5  =  0,  (|,  |). 

20.  a; -22/ +12  =  0,  (12,  12). 


20  ANALYTIC    GEOMETRY. 


21.   3/  =  a;(±>/2-l)  +  4(±V2-l).  22.   — ^-^ -- 

24.  One  of  the  points  of  contact  is  (—1,  11).  The  vertex  of  the  para- 
bola is  the  point  (—  9,  3) ;  therefore  the  distance  from  the  vertex  to  the 
intersection  of  the  axis  and  the  ordinate  of  the  point  of  contact  is 
equal  to  8;  therefore  the  subtangent  =  10;  therefore  a  second  point  of 
the  tangent  is  (—17,  3) ;  therefore  the  equation  of  the  tangent  through 
(-1,  11)  is  a;-2y  +  23  =  0. 

25.  j  (ii.)   a-ix=      2p{y  +  y{), 
l(iii.)  x^x  =  -2p{y  +y^). 

Ex.  30.      Page  131. 


1. 

3/2  =  24  a; -144. 

2. 

f 

=  16x. 

3.   3/2  =  _17rc. 

4. 

f  22/2-ll,r  +  12,y +  7 
l2a;2+lLr  +  12y-3 

■3  = 

-0; 
-0. 

or 

5.  (y+7y  =  4:{x-3). 

6.  3y2  =  4x. 

7. 

3a'2  =  4y. 

8. 

h 

8x  +  3 

i  =  0,  8x±15y-3  =  0. 

10 

■  4  on  OX; 

8  and -2  on  OY. 
4(2  +  \/3)p. 
^-(1.)     y=-x  +  2, 
\{n.)    2V2, 
l(iii.)  a:  +  y-6  =  0. 

20. 

3/2  =  9a:. 

J.V. 

11. 

12. 

21. 
22. 

23. 

3/2  =  8. T. 

3/2  =  4^^-^. 

3/2  =  -.. 

13. 

x  +  y-Q  =  0. 
2p 

y-yi  =  -^i^-^i)- 

24. 

V/l2  +   P 

14. 

25 

3/2=2(2r-s)a;. 

15. 
16. 

(8,  4),  (2,  10). 
(2,  4),  (11,  10). 

26. 

27. 

28. 

4pV2. 

The  equation  of  the  circle  is 

(.T-  3)2  +  (3/  --1)2  =2^. 

(-3i>,0). 

17. 

h±\/h-^-iap 

18. 

f  A  left-handed  A'-parab( 
j  Latus  rectum  =  —.2. 
i  Vortex,  (-2,  0). 
Focus,  (-  I,  0). 

3la. 

29. 
30. 

j(i'.  ±2p); 
1  45°  and  135°. 

[  Directrix,  x=  —  |. 

31. 

4p2. 

19. 

4  a. 

34 

The  parabola  y'^  =  px. 

35. 

y'  =  2px-p\ 

36. 

f  =  2px-2p\ 

37. 

y-  =  -p^- 

38. 

y2  =  2px  +  2f. 

43. 

Take   the   given   line   as 

ANSWERS.  ;21 

The  loci  in  exercises  35-38  are  parabolas,  the  latus  rectum  in  each 
being  half  that  of  the  given  parabola.  If  the  given  parabola  is  y^  =  ipx, 
the  equations  of  the  loci  are  : 

39.  The  straight  line  y  =pk. 

40.  The  parabola  y*  —  4pa;  =2>2^^ 

41.  The  straight  line  kx=p. 

42.  The  parabola  {^—pY+y'^—ri- 

the  axis  of  y,  and  a  perpendicular 
through  the  given  point  as  the  axis  of  x,  and  let  the  distance  from  the 
point  to  the  line  =  a.     The  locus  is  the  parabola  y^  =  2al  x ]• 

Ex.  31.     Page  143. 
10.    3x-5y-6  =  0.  ^2ay==ipx. 

11     8v— '^5  =  0  14    ]  The  chord  is  parallel  to  the 

tangent  at  the  end  of  the 

12.  13a:  +  22y  +  ^  =  0.  I      diameter. 

13.  x-y-l=0.  15.   3/2  =  52a\ 

.20.  If  the  equation  of  the  given  parabola  is  y^  =  4ipx,  the  locus  is 
the  parabola  y'^  =  p{x  —p). 

21.  If  the  equation  of  the  given  parabola  is  y^  =,  4^^.^  the  locus  is 
the  parabola  y'^  =  p{x  —  3p). 

22.  Take  the  given  line  as  the  axis  of  y,  and  a  perpendicular  through 
the  centre  of  the  given  circle  as  the  axis  of  x.  Let  the  radius  of  the 
circle  =  r ;  distance  from  the  centre  to  the  given  line  =  a.  There  are 
two  cases  to  consider,  since  the  circles  may  touch  the  given  circle  either 
externally  or  internally.     The  two  loci  are  the  parabolas 

2/2  =  2(o  +  r)a;+r2-a2, 
3/2  =  2(a-r).T  +  r2_a2. 

23.  Let  2a  be  the  given  base,  ab  the  given  area;  take  the  base  as 
axis  of  X,  its  middle  point  as  origin  ;  then  the  locus  is  the  parabola 

x^  +  by  =  a^. 

Ex.  32.      Page  153. 
1.   5,  4,  3,  f.  2.    V2,  ],  1,  VI.  3.   2,  VS,  Vl,  iVl. 

4.  J-.  J-.  .'AE^.  ^V^^- 


13. 

a;2  +  2y2  =  l00. 

14. 

8a;2  +  9y2=8a2. 

15. 

2:V3. 

16 

ah 
X  —  V  —  -\ 

Va'^  +  h' 

18. 

a.f),(-f,-§)- 

22  ANALYTIC    GEOMETRY. 

5.  «\/6. 

6.  c  =  ^\/3. 

7.  4x2  +  97/2  =  144. 

8.  250-2+169^2  =  4225. 

9.  144x2  +  2251/2  =  32,400. 

10.  16x2  +  252/2  =  1600. 

11.  25x2  +  1692/2  =  4225.  19.   (1,  2),  (1,  -  2). 

12.  115x2 +  252y2  =  4140.  20.   (3, 1),  (3, -1),  (_3, 1),  (-3, - 

21.  See  No.  11.     The  equation  of  the  locus  is  x2  +  2?/2  =  r-. 

22.  Taking  as  axes  the  two  fixed  lines,  and  putting  AP==a,  BP-- 
the  acute  angle  between  AB  and  the  axis  of  x  =  cp,  we  find  that 

X  =  a  cos  0,     y  =  &  sin  (}>. 
Therefore  P  describes  an  ellipse  whose  equation  is 

e!  +  ^  =  i. 


a* 


A 
23.    The  two  straight  lines  y  =  ±— x.     The  locus  is  imaginary  when 

the  values  of  y  are  imaginary  ;  that  is,  when  A  and  B  have  unlike  .«igns. 
29.    The  equations  of  the  sides  are 

ah  ah 

X  =  ±  — -————,     y  = -— — -  , 

Va2  +  62  Va2  +  62 

4a262 


area 


62 


Ex.  33.     Page  160. 


4x±9y  =  35, 
9  X  ±  4  y  =  6. 
2x±3yV3-12  =  0,_ 
6xV3±4y+9V3+2V6  =  0. 
4x+    y  =  10, 
x-4y +  6  =  0; 


5.   :i-+^  =  l 


m 


2  ,,2 


6.   9x2  + 25  y' 


7. 

2y  =  x±10. 

8. 

4x-3y  ±  Vl07  =  0. 

9. 

«2                            62 

x  =  ± ,    ?/  =  + 

Va2+62              Va2+62 

10. 

Same  answers  as  No.  9. 

11. 

62  :  a\ 

12. 

x  =  ±-«,    y  =  ±A. 
V2                V2 

13. 

y  =  4,    3x  +  34y=17. 

ANSWERS.  23 

14.  Tlio  equation  ±  Vox  ±  3?/  =  9a  represents  the  four  tangents. 

15.  a  Vl  —  e^  cos"''  <p.  16.    ^  (a^  tan  <^  +  6"^  cot  <(>). 
17.    The  extremities  of  the  latera  recta. 

19.    The  method  of  solving  this  question  is  similar  to  that  employed 
in  \  147.     The  required  locus  is  the  auxiliary  circle  x^  -Vy^  =  a?. 


1. 

2. 

x  =  8,  40y  = 

1  -¥-.  ¥• 

Within. 
1 

Va' 

1 

V2' 

=  9i 

Ex. 

C  +  72: 

=  0. 

34. 

Pag( 
8. 
9. 

10. 

11. 
12. 

13. 

14. 
15. 

s  161. 

hx  +  ay  =  ab-\/~l. 

X               y    . 

-  cosd)  +  y  sin  d)  =  1. 

a       ^      b       ^ 

ah 

3. 

Va^-eV 

4. 

oVl  —  e^  cos'^'c^. 
e^ :  h\ 

5. 

Vl3±l 

2V3 

e%y^ 
aW 

6. 

a;  ±  2/  ±  Va"''  + 
hx  +  c_y  =  6c  -y 

6-'': 

V(l-e2)(a2-e'^Xi2). 

7. 

/2q 

,2     ,     Z.-i 

tan0    V^-^ 

18.    The  locus  is  the  minor  axis  produced. 


19.  The  ellipse   [  x  —  -\  +  2^  =  —  ;  centre  is  (  -,  0  )  ;  semiaxes  are  - 
,  V        2/       4      4  V2      /  2 

and  r.  ^  '  ^        '' 

20.  The  ellipse  a^{y \-\-h'^x^  = ;  centre  is  [  0, -]  ;    semiaxes 

a        1  6  \         -'/  V     ^/ 

are  -  and  ~ 
2  2 

In  21-23  take  the  base  of  the  triangle  as  the  axis  of  re,  and  the  origin 

at  its  middle  point. 

21.  The  ellipse  (s^  -  e-)  x'-  +  sh/  =  s'is''  -  c'). 

22.  The  ellipse  kx"^  +  2/^  =  ^C^- 

23.  The  circle  {x  +  cf  +  y^  =  4a2. 

Ex.  35.     Page  176. 
1.    a;  =  ±-  =  ±--  3.    20a; +  63y- 36  =  0. 

.     /     Aa"         Bh^\  Q    /•^     2      &'     /••^     2      *     /•••x     2      1 


24-  ANALYTIC    GEOMETRY. 


W44f 


9.   a^l\'-'  '"'"^  ^     b  =  lVT 


10.    3  .r  +  8  y  =  4,    2  a;  -  3  y  =  0.         11.    a^yja;  =  Rrj?/. 

12.  Area  =  —  (7?2  +  n),  m  and  7i   being  the  two    segments  (use  th 

2a 
polar  equation). 

13.  26a; +  33?/ -92  =  0.  14.    .r  +  2y  =  8. 
15.    h^x  +  a^?/  =  0,  b\x  -  ahj  =  0,  cv^y  +  Z^li-  =  0. 


16. 

Z>,T  +  ay  = 

0. 

26. 

See  I  156. 

17. 

See  I  157. 

27. 

See  I  159. 

18. 
23. 

See  g  158. 
a     b 

30. 
31. 

1   f  cc  Q 

24. 

y^d'  -  P 

xVP- 

-b\ 

b'- 

e  =  ^\/ti. 

32. 

25. 

^      1  -  e2  cos2  d 

34.  16a;2  +  49^/2  _  128a:  -  6863/  +  1473  =  0. 

35.  2a  =  18,  2i  =  10.  36.    --^  +  y'^  =  bx. 

144       "^ 

37.   Centre  is  (-1,  1),  2a  =  2,  26  =  4. 
.  38.    Centre  is  (2,  4),  2a  =  8,  26  =  6. 
39.    Centre  is  (0,  -  J),  2a  =  3,  26  =  1. 


40.   cos  (/)  =  a/4— r!-  41.    tan  0  =  — 

»  /-/2  —  /i2 


6 


42.  Find  the  ratio  of  y^  to  the  intercept  on  the  axis  of  3/ 

43.  b^hx  +  a^ky  =  bVi""  +  a''k\  .-     t^,       ,,.         a;^      y^      , 

'^  47.    The  elapse ^  -^  =  J 

a     , 
pse  -  + 


46.    The  ellipse  i!i!  +  ^  =  2 


62  48.    The  ellipse  62.1-2  +  a2y2  =  b^c 

49.    The  ellipse  25a;2  +  I63/2  -  48y  =  64. 

Ex.  36.     Page  186. 


yi        .,2 


1    i^_2C==i  3.    8.x2_3/2_Sa2. 

64     49 
2  2  4.    625a:2-84y2  =  10,000. 

25      144  5.    2.T2_2y2  =  c2. 

7.    a  =  4,  6  =  3,  c=»5,  c  =  f.  latus  rectum  ==  |. 


ANSWERS.  25 

8.  IGy"^  —  dx^^  25,  transverse  axis  =  6,  conjugate  axis  =  8,  distance 
between  focus  =  10,  latus  rectum  =  y. 

9.  a  :  6  =  1  :  V3.  11.   e  =  VI 

14.    Foci,  (5,  0),  (—5,  0);  asymptotes,  y  =  ±^x.  17.   b. 

Ex.  37.     Page  188. 

1.    16  a:  -  97/  =  28,  9  .r  +  I63/  =  100,  f ,  -%\  3.   x"  -2/^  =  9,  (5,  4). 

4.   The  four  points  represented  by 

±  a'  ±  &2 

X  =  ■!       7/ = • 

Va-'  -  ^2  Va''  -  h' 

If  J'^  >  a^  the  points  are  imaginary.     If  the  hyperbola  is  equilateral, 
the  points  are  at  an  infinite  distance. 

9.    ^.  10.   ^--^'  =  1. 


V3 


m'      n 


11.   "When  a  is  less  than  h.  12.  The  circle  a;^  +  y^  =  ^2^ 

Ex.  38.  Page  189. 

1.  2he,  ae^.  11.  y  =  ±x's/2  +  a. 

2.  U  and  6.  12.  (0,  ±  Va^  _  6'^). 

3.  The  sum  =  2  car.  13.  6^  >  ^2. 

8.    (a,  6 \/2),  (a,  -  6 \/2).  14.  64  a;  -  9 y  +  741  =  0. 

10.    They  are  equal.  15.  ?/ =  4  a;  ±  8  \/2. 
,2                             52 


16.   a;  =  ± 


Va2  -  4  i-^  Va2-462 

Ex.  39.     Page  201. 
1.   9a;  +  12y  +  16  =  0.  6.    a. 


2.    X 


a 


8.    75.r-167/=0. 


<^  9.  245.^-123/ -1189  =  0. 

3.  ^.  10.  |V3. 

^  17.  See  §  150. 

4.  (-^^    f  \  18.  4.ry  =  -(a^  +  6^). 

•^  X       y 

5.  a-  +  a  =  0.  19-  F,  +  y-,=-- 


26  ANALYTIC    GEOMETRY. 


a)    ^_^_^'  =  o  22.  p=«^(^l::ill 

(ii.)  ^-^'+?5  =  o.  23.    -^  ^'' 


a-      b'^       a  '    *^       e^co^^d-l 

24.  The  equation  represents  an  equilateral  hyperbola,  with  it?  trans- 
verse (real)  axis  parallel  to  the  axis  of  y.  The  centre  is  (1,  —  2) ;  the 
semiaxes  are  each  equal  to  2. 

25.  The  hyperbola  Zx^-y"^  +  20 x-  100  =  0.  The  centre  is  the  point 
(— J3O-,  0).    Changing  the  origin  to  the  centre,  we  obtain  9a;2  — 33/2.=  400. 

26.  The  locus  is  the  curve  2a;y  —  7  a;  +  4?/ =  0.  If  we  change  the 
origin  .0  the  point  (A,  h),  we  c*an  so  choose  the  values  of  h  and  k  as  to 
get  rid  of  the  terms  containing  x  and  y.     Making  the  change,  we  obtain 

2xy  +  (2k-7)x  +  {2h  +  A)y-7h  +  ik  +  hk  =  0. 
If  we  choose  h  and  k  so  that  2  A +  4  =  0,  and  2  A;  — 7=0,  that  is,  if 
we  take  A  =  —  2,  ^  =  |,  the  terms  containing  x  and  y  vani^,  and  the 
equation  becomes  2xy  =  7.  Hence  we  see  (§  194)  that  the  locus  is  an 
equilateral  hyperbola,  and  that  the  axes  of  co-ordinates  are  now  the 
asymptotes. 

27.  The  equilateral  hyperbola  2xy  =  a^. 

28.  Taking  the  base  as  axis  of  x,  and  the  vertex  of  the  smaller  angle 
as  origin,  the  loci  are  the  axis  of  x  and  the  hyperbola  3x^  —  y^  —  2ax  =  0. 

Ex.  40.     Page  221. 

1.  The  ellipse  72 a:'  +  48 y^  =  35.  9.  The  ellipse  4.^2  +  9y^=  36. 

2.  The  parabola  2/^^  =  —  ^.^.  10.  The  parabola  3/^  =  2a;. 

3.  The  parabola  y^  =  2.rV2.  11.  The  ellipse  4a'2  +  9^/2  =  36. 

4.  The  ellipse  4^2  +  2/  =  1.  12.  The  ellipse  4a;2  4-  y'^  =  100. 

5.  The  hyperbola  32^2-48/ =  9.  13.  The  hyperbola  4 a;^- 9 7/^=  36. 

6.  The  parabola  _y 2  =  3,^.^0  -^^  The  straight  lines  3/=.r,y=  — 5. 

7.  The  ellipse  9x2  +  3  7/2  =  32.  15.  The  parabola  7/2  =  5  a;. 

8.  The  hyperbola  4x2-4y2^1_Q  jg  The  parabola  25a;2+2j/V5  =  0. 


r: 


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